The syntactic lexical game ′ 1 is shown in Figure 7.3. In it, the only initial situation is ′ 1 . This corresponds to the initial situations 1 and 1 ′ in Figure 7.2. If there had been an ambiguity in the parse of 1, say another tree ′ 1 , there would have been a second initial situation ′ 1 ′ in Figure 7.3. Notice where the primes are placed, whether on the symbol itself or on the subscript 1 or both. The prime on the symbol corresponds to the syntactic game and the prime on the subscript determines whether it is an alternative initial situation in the relevant game, semantic or syntactic.

In this game, there is no intention to convey the parse 1 in ′ 1 as there was in the previous game 1. In semantic games, there is an intention whether explicit or implicit. In syntactic games, neither the speaker nor the addressee are likely to be aware of the parse of the corresponding word or expression. All that happens is the initial situation is a part of the utterance situation which includes a fact that the relevant parse tree is being transmitted or conveyed, just like a transfer of information from one hard disk to another.

The rest of the diagram should be straightforward to read. A has just the one choice of utterance 1 and B has just one choice of syntactic interpretation 1. The payoffs received are the same as before because the interpretation is obviously correct with respect to what is being transmitted in ′ 1 .

The prior probability ′ 1 = (1 ∣ 1, 2, 2; ) where the 1 is either 1 or ′ 1 , 2 is either 2 or ′ 2 as before, and 2 is just 2 as there is no other parse tree corresponding to 2. It is also influenced by the parameter . Since there is just one alternative 1 in this game, ′ 1 = (1 ∣ 1, 2, 2; ) = 1 independent of what values the conditioning variables adopt.

This syntactic lexical game ′ 1 is obtained by applying the syntactic game map ′ to the word 1.That is, ′ (1) = ′ 1 . Unlike the case of the semantic game 1, ′ 1 is constructed entirely nonconsciously or subpersonally by A. This is also often

### 7 Defining Communication Games

true for semantic games but in those instances there may be a kind of semiconsciousness present. The syntactic game ′ 1 uses the parsing alternatives provided by the Syntactic Constraint and other information analogous to the information used in constructing 1. That is, the map ′ is available to A just as was and all that is required is for A to apply it to 1.

This game also presents a choice situation for both agents, as yet only in A's head. All the choices on both sides – the utterance and the prior probability on the speaker's side and the interpretation and the prior probability on the addressee's side – are trivial. So, the solution to this game is also trivial but it still does depend in an otiose way on the values of the conditioning variables 1, 2, and 2 and so on the solutions to those games just as the solutions to those games depend on the solution to ′ 1 .

Now, we move to the next set of lexical games for the second word 2 = ran shown in Figure 7.4 and Figure 7.5. These diagrams should be relatively straightforward to understand.

Figure 7.4: Semantic lexical game 2

In 2 in Figure 7.4, the speaker is conveying 2, although because 2 is semantically ambiguous, the addressee could potentially interpret the word as conveying ′ 2 as well. This choice is represented by two branches labeled 2 and ′ 2 emanating from the upper intermediate situation in the oval. The payoffs follow the same pattern as in 1. In 2 ′ in the lower half of the same game, a similar choice

### 7.3 The Generation Game

presents itself to the addressee but this time the payoffs are reversed because in this hypothetical situation A is conveying the other possible meaning ′ 2 . Again, the primed payoffs are greater than the corresponding primed payoffs.

The prior probabilities for 2 are 2 = (2 ∣ 1, 1, 2; ) and 2 ′ = ( ′ 2 ∣ 1, 1, 2; ) and these must sum to 1 as there are only two possible meanings the speaker could be conveying. Since these probabilities are also functions of the conditioning variables and parameter , there are different *versions* of essentially the same game 2 when 1 takes on different values as 1 and 2 are just 1 and 2 and is held fixed.

As I said above with 1, (2 ∣ 1, 1, 2; ) > ( ′ 2 ∣ 1, 1, 2; ) because the presence of 1 is compatible with 2 as Bill Smith is more likely to be running in an election than running in a race. This is also reinforced by as the topic or goal of conversation is the local election. On the other hand, (2 ∣ ′ 1 , 1, 2; ) < ( ′ 2 ∣ ′ 1 , 1, 2; ) because now 2 and ′ 1 are incompatible as one involves running in an election and the other involves Bill Jones whereas ′ 2 and ′ 1 are compatible because Bill Jones is, in fact, a runner. This is so despite the presence of which might bolster the former probability because incompatible meanings are much worse than an irrelevant statement. Once again, the parse trees do not play any role except to confirm all the probabilities as they are unique. If we assume the same numerical values as before, both players could be choosing between (2, 2 ′) = (5/6, 1/6), one version of 2, and (2, 2 ′) = (1/4, 3/4), another version of 2. Again, the actual numbers themselves are determined by a variety of objective and subjective factors that are exogenous but the choice between them is endogenous.

The same kinds of choices and payoffs occur in Figure 7.5 except that this time a trivial syntactic ambiguity has to be "resolved" as in ′ 1 . This time the only prior conditional probability is ′ 2 = (2 ∣ 1, 2, 1; ) = 1 as before.

The games 2 and ′ 2 are obtained by applying the same semantic and syntactic maps and ′ to 2. That is, (2) = 2 and ′ (2) = ′ 2 . Again, as argued above, both maps are available to A and so the relevant games can be constructed without any further assumptions.

This completes the discussion of semantic and syntactic lexical games as constructed nonconsciously by A. There are two semantic games and two syntactic games, each corresponding to the utterance of 1 or 2 in . Moreover, as I have shown in detail, the games are *interdependent* because the solution to each of them depends on the solutions to the others via the prior probabilities which are conditioned by these solutions. In this example, the syntactic games were trivial and solvable immediately and so their dependence on other games was

### 7 Defining Communication Games

redundant. Later in Chapter 9, I will look at a more complex example with syntactic ambiguities as well. Because in general one cannot know which version of a semantic or syntactic game is going to be optimal without solving all the other lexical games, they have to be solved together. In this sense, these interlocking games are circular not only because each individual game involves an interdependence between speaker and addressee but also because no game can be solved without solving all the games simultaneously.

The next step is to consider the *products* of these lexical games to form phrasal and sentential games. In the very simple two-word sentence at hand, there are no phrasal games because we get the sentential games at once. I show the semantic sentential game 12 = 1 ⊗ 2 in Figure 7.6.

There are several new things in this game product. First, the tree diagram is a "product" of the trees of 1 and 2. 7 Next, there are four initial situations 12 = 1 ∪ 2, 12′ = 1 ∪ 2 ′, 1 ′2 = 1 ′ ∪ 2, and 1 ′2 ′ = 1 ′ ∪ 2 ′. The latter situations 1, 1 ′, and 2, 2 ′ are the initial situations of the two semantic games 1 and 2 that are being multiplied. That is, the initial situations of the product are derived from the initial situations of the multiplicands as they should be and 2 x 2 = 4 such combinations are possible from the two initial situations of 1 and the two initial situations of 2.

I have abbreviated the sentence = 1 ○ 2 as 12 and the products 1 ⊙ 2, 1 ⊙ ′ 2 , ′ 1 ⊙ 2, and ′ 1 ⊙ ′ 2 as 12, 1 ′ 2 , ′ 12, and ′ 1 ′ 2 , respectively. The speaker's single choice of utterance 12 is a concatenation of the individual utterances 1 and 2 from 1 and 2. And the addressee's possible interpretations are 12, 1 ′ 2 , ′ 12, and ′ 1 ′ 2 , where again the components are obtained from the respective games. The payoffs corresponding to the relevant branches in the multiplicands are added to give the payoffs in the relevant branch of the product. For example, the payoff for A in the topmost branch corresponding to the path 12, 12, 12 is A +A, which is the sum of the A in 1 corresponding to the path 1, 1, 1 and the A in 2 corresponding to the path 2, 2, 2.

Finally, the four priors are 12 = (1, 2 ∣ 1, 2; ), 12′ = (1, ′ 2 ∣ 1, 2; ), 1 ′2 = ( ′ 1 , 2 ∣ 1, 2; ), and 1 ′2 ′ = ( ′ 1 , ′ 2 ∣ 1, 2; ) which must sum to 1 as they represent all the alternatives. Again, it should be easy to see how these probabilities are generated from the corresponding probabilities in the multiplicands 1 and 2 where the priors are 1 = (1 ∣ 2, 1, 2; ), 1 ′ = ( ′ 1 ∣ 2, 1, 2; ) and 2 = (2 ∣ 1, 1, 2; ), 2 ′ = ( ′ 2 ∣ 1, 1, 2; ). It is clear that 12 > 1 ′2 ′ because both are conditioned by 1, 2, and and because is about the local

7This game tree product is, of course, different from the parse tree product described in Section 4.2.

Figure 7.6: Semantic sentential game 12 = 1 ⊗ 2

### 7 Defining Communication Games

election it pushes up the former and pushes down the latter. The parse trees 1 and 2 are both unique and play no role. The "cross" probabilities 12′ and 1 ′2 are both very much lower than the other two because they represent incompatible interpretations, either Bill Smith the politician running in a race or Bill Jones the runner running in an election. So 12 is the highest among these four prior probabilities. A possible set of values could be (12, 12′, 1 ′2, 1 ′2 ′) = (0.5, 0.1, 0.1, 0.3). I will presently be showing how all these games are to be solved but I point out here that the prior probabilities play a key role.

The product game 12 = 1 ⊗ 2 can also be obtained directly by applying the semantic map to the sentence 1 ○ 2. When this is done, it becomes evident that 12 ≡ (1 ○ 2) = (1) ⊗ (2) ≡ 1 ⊗ 2.

The syntactic sentential game ′ 12 = ′ 1 ⊗ ′ 2 is shown in Figure 7.7.

$$\begin{array}{c} \begin{array}{c} s\_{12}' \\ \hline \end{array} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} t\_{1} \end{array} \end{array} \begin{array}{c} t\_{1} \end{array} \begin{array}{c} \\ \begin{array}{c} \\ \end{array} \end{array} \begin{array}{c} \\ \end{array} \begin{array}{c} a\_{\mathcal{A}} + a\_{\mathcal{A}}, \text{ ag} + a \text{g} \end{array} \end{array} \end{array} \begin{array}{c} \begin{array}{c} a\_{\mathcal{A}} + a\_{\mathcal{A}}, \text{ ag} + a \text{g} \end{array} \end{array} \end{array}$$

Each multiplicand has just one initial situation ′ 1 or ′ 2 and so there is just 1 = 1 × 1 initial situation ′ 12 = ′ 1 ∪ ′ 2 in the product. The interpretation 12 = 1 ⋆ 2 = 12 as shown earlier. The payoffs are the sum of the corresponding payoffs in the component games as before. The prior probability ′ 12 = (1, 2 ∣ 1, 2; ) = 1 as the reader should now be able to work out. This is a trivial game and even though the prior probability is conditioned by other variables that take on different values depending on the optimal solution to the semantic games 1, 2, and 12, it is always just 1 so the game is solvable immediately. As with the semantic sentential game, this syntactic product can be obtained directly by applying the syntactic map ′ to 1 ○ 2 and so we have ′ 12 ≡ ′ (1 ○ 2) = ′ (1) ⊗ ′ ′ (2) ≡ ′ 1 ⊗ ′ ′ 2 .

This is the Flow Constraint from the speaker's viewpoint. Recall that all the interlocking games constructed so far – 1, 2, 12 on the semantic side and ′ 1 , ′ 2 , ′ 12 on the syntactic side – actually carry a superscript A to mark this viewpoint. These are the games generated by the speaker when he considers the sentence = 12 = bill ran as a possible thing to utter in order to convey 12 = *Bill Smith stood (for election)* to B and thereby get across the whole content = *Bill Smith ran in the local election* in order to evoke the response , acceptance, by B. This content 12, if indeed it is the content of an utterance of in , something I have yet to show, would be called the *locutionary meaning* of the utterance because it comes directly from the words uttered in the situation . As

### 7.3 The Generation Game

I have said before, the sentence has no meaning without a context and does not have a conventional meaning at all. As the Semantic Constraint makes clear, only words have conventional meanings and they, together with the situation , yield the locutionary meaning and, indeed, also the illocutionary meaning of the utterance which is the meaning that comes largely from the context and only indirectly from the words uttered. This latter meaning, which includes things like free enrichment, modulation, figures of speech, implicature, and illocutionary force, will be addressed in Part IV.

I will postpone showing how these six locutionary games are solved because I first want to develop the complete model of locutionary communication. For the moment, just assume that their joint solution is in fact 12 rather than 1 ′ 2 , ′ 12, or ′ 1 ′ 2 , the other possibilities shown in Figure 7.6. I will label this locutionary meaning ℓ = 12 = *Bill Smith stood (for election)*. The full illocutionary meaning will be labeled so that the complete meaning of the utterance will be = ℓ ⊙ = *Bill Smith ran in the local election*.

On the other hand, it is obvious that the equilibrium or optimal parse of the utterance is just = 12 = 12 as indicated in Figure 7.7 as it is the only possibility. In more complex cases where syntactic ambiguities are present, the solution will be correspondingly more complex and syntax will also play a more active role in the overall determination of both locutionary and illocutionary contents and vice versa. Indeed, the syntactic and semantic (and phonetic) games all have to be solved together because they generally cannot be solved "locally" except when they are trivial. While the solution process involves a slightly elaborate description, it involves rather easy probabilistic computations that are straightforward to implement in our nonconscious (and semiconscious) neural processes.

The six interdependent *local* games that have been constructed so far can also be collected into a locutionary global game () = {1, 2, 12, ′ 1 , ′ 2 , ′ 12} and we can say that when A considers how B might interpret an utterance of in in his head he will construct (). This interdependence of the games of partial information is another reason why they are not games of incomplete information (or signaling games). If is a solution function that maps games into their solutions, we can also write, based on our assumption that the joint solution to the semantic locutionary games is 12 and to the syntactic locutionary games is 12, that [()] = {(1), (2), (12), ( ′ 1 ), ( ′ 2 ), ( ′ 12)} = {1, 2, 12, 1, 2, 12}. In other words, the solution to each local game, whether semantic or syntactic, is compatible with the solutions to all the other games because they must agree not only in the contents they deliver but also in the values of the conditioning variables that get set in the prior probability functions. For example, in the first lexical game 1, the prior probability function

### 7 Defining Communication Games

1 = (1 ∣ 2, 1, 2; ) = (1 ∣ 2, 1, 2; ) *in equilibrium*, which shows that the conditioning variables are 2, 1, 2 which all agree with the solutions to the other five games. Further, the equilibrium paths that will actually be followed in these six games will all begin with the actual situations, 1, 2, 12 in the three semantic games and ′ 1 , ′ 2 , ′ 12 in the syntactic games, because all the other initial situations are just counterfactual possibilities. And these paths will follow the branches that are consonant with the solutions listed above. If these paths are traced in all the trees then the payoffs actually delivered to the speaker would be A, A, A + A in the three semantic games and A, A, A + A in the three syntactic games. We can just take the maximum value A + A of all these six numbers as the payoff that would be actually delivered to the speaker if he were to utter in . Precisely what we do is somewhat arbitrary: the payoffs could also be added up to yield 8A or just the lexical payoffs could be added to yield 4A. This is because of the affine nature of utilities mentioned in footnote 9 in Section 5.1.

There is one important matter that remains to be specified with respect to the products of games. I considered the products of two semantic games and of two syntactic games but why can't one multiply a semantic game with a syntactic game? After all, both yield *contents* and I have been saying that semantic and syntactic contents are to be treated uniformly. The answer is one can – by forming ordered pairs of possible semantic and syntactic contents. For example, the "pure" semantic game 1 in Figure 7.2 would have (1, ) in place of just 1 and ( ′ 1 , ) in place of just ′ 1 where is just the empty tree. Likewise, the "pure" syntactic game ′ 1 in Figure 7.3 would have (**1**, 1) in place of just 1 where **1** is just empty or no information. Then we can perform the product 1 ⊗ ′ 1 = 11′ which would have (1, 1) and ( ′ 1 , 1) as the interpretive choices for B. This shows that the locutionary global game actually contains many more local games. Indeed, it is not enough to consider just ordered *pairs* of contents but we need 2-tuples where is the number of words in the sentence. This is because we can also form products like 1 ⊗ ′ 2 = 12′ as well as 12 ⊗ ′ 12 = 121′2 ′, the full mixed sentential product.

I will not bother to list all of them as there are too many permutations of the indices. In fact, the upper bound for the number of these games is 2 2 − 1 = 15 when = 2. 8 So one can see that the number of games grows quite rapidly

8 It is instructive to work out this formula to check one's understanding of these games. In general, there will be 2 indices, 1, 2, 3, …, for semantic lexical games and and 1 ′ , 2 ′ , 3 ′ , …, ′ for syntactic lexical games. These indices form a set of 2 elements which has 2 2 subsets. And as there is no null subset involved, we subtract 1, giving us the formula 2 2 −1. The reason it is an upper bound is that there will be many zero products and the corresponding games would be discarded.

### 7.3 The Generation Game

with , the length of the sentence. I will show in the next chapter that the only product of semantic and syntactic games that matters is the full mixed sentential product so I will ignore the other mixed products. Remarkably, the only games needed to understand an utterance turn out to be the pure lexical games, a vast computational simplification resulting from our rationality.

## **7.3.4 Back to the Generation Game**

Thus, our conclusion so far is that, based on the Content Selection Game induced by the Setting Game in , A resolves to convey in in order to evoke the response from B, and then, in the Generation Game that is induced by the Content Selection Game, casts about for a sentence that would do the job, finds as one such sentence by some assumed conversion process from to , constructs () in his head as described by the Phonetic, Syntactic, Semantic, and Flow Constraints, solves it with the map , checks that the equilibrium content is compatible with , and receives a value A [()] = A + A. We say the *value* of () to A is A + A. This entire computation, or some approximate version of it, presumably takes only milliseconds to perform in a speaker's head and happens nonconsciously. In actual fact, it is not just () that is constructed but also (), the global set of local illocutionary games corresponding to the illocutionary meanings of the utterance, whose description I have postponed to Part IV. In other words, it is the global game () = () ∪ () that the speaker identifies when he contemplates uttering and it is the global game's value A [()] that he computes. I realize this is a lot of terminology but once we have set it up, the discussion will flow very smoothly.

There is one other factor that enters into A's consideration of . That is the *cost* of uttering , A (), which is based on a variety of factors, mostly objective ones such as the length and complexity of the sentence and the effort required to mentally process it and to physically produce it but also subjective ones especially related to the consequences of making certain information explicit and leaving certain information implicit. These subjective costs have to do partly with aesthetic things like conversational style but also with how making certain things explicit may affect the relationship with the addressee in positive or negative ways depending on whether these explicit things are polite or impolite and how leaving certain things implicit may result in costly errors. The mental processing involves playing the global game and determining its value. The *net* value A would receive from uttering is then A () = A [()] − A (). 9 This

9Here, I have used the same symbol A for a function of and for a function of () to avoid multiplying symbols needlessly. Also, note that this symbol is different from A, the payoff to A shown in the Content Selection Game in Figure 7.1.

### 7 Defining Communication Games

finally completes my description of what happens when A imagines uttering in . Essentially, he computes its net value A () based on the value A [()] of the global game determined by the four Constraints and its cost A (). I will consider the details of costs in Section 8.4.

In the same way, A may compute the net values of other sentences he could utter in to convey , possibly ′ = bill smith ran and ″ = bill ran in the election. These net values would be determined by the global games ( ′ ), ( ″) and the associated costs A ( ′ ), A ( ″) of the two sentences. Each of these global games would involve the corresponding lexical, phrasal, and sentential semantic and syntactic speaker games (as well as the relevant illocutionary games) and their global solutions.

On the other hand, the limits of rationality mentioned earlier may make him abandon his search for more sentences if he feels sufficiently satisfied with the best net value he has computed so far. This is a trial and error process as becomes especially clear when one is writing rather than speaking as then there is ample scope to try out different sentences to express roughly the same content. Indeed, when writing is considered, it becomes obvious that there is an interdependence between the Content Selection Game and the Generation Game because a new sentence with a slightly improved content may suggest itself in the Generation Game and then the new content would have to be evaluated in an altered Content Selection Game.

In any case, he considers one or more sentences and this leads to the choice he faces in the Generation Game () as shown in Figure 7.8.

Figure 7.8: Partial Generation Game ()

This may appear superficially like a one-person decision problem but keep in mind that the net values are based on the values of the corresponding global games which involve both A and B. () is so far just in A's head. His solving it involves choosing the sentence with the highest net value and, finally, publicly uttering it in .

### 7.3 The Generation Game

I will assume is in fact the best sentence to utter in because it conveys the same content as the other sentences with a lower cost. Its cost is lower because it is shorter and simpler than the other sentences and the physical and mental effort involved is therefore lower (even though there may not be any illocutionary games required for the other sentences because all the information has been made explicit in them). Also, if the ongoing conversation is about the local election, then aesthetic considerations might imply that repeating the information about the election explicitly is inelegant. For example, if B has just referred to the election explicitly in her prior utterance to A, then the election is already very prominent in the context. These sorts of things lead to the use of pronouns, for instance. There may be no very weighty relationship-related dimensions involved and no politeness considerations in this kind of purely informational and casual exchange. I will consider such psychosocial factors in Sections 8.6 and 18.2 and Chapter 11 as the first task is to lay out the model in its simplest form.

I end this section by reiterating the point made in Section 3.1 that sharply separating the Content Selection Game from the Generation Game is just a useful idealization. The two games are interdependent because identifying the optimal content to convey leads to the search for the optimal sentence to utter and vice versa. For example, in considering (), A may opt for something more tentative and less assertive such as = i think bill ran based on his own and his addressee's psychology as well as on their interaction so far. This would take him back to an altered Content Selection Game with a slightly different content corresponding to and so on. And, indeed, bits of content and bits of language rather than entire sentences may go back and forth between the two games with alterations in both games as new possibilities arise. Speaking and writing are ultimately creative acts so there can be no complete closure to this process. That is part of the reason why I do not as yet have a model of how sentences are generated from contents: the map is partly open-ended as it depends on many factors including the speaker's mastery of the language.

To the best of my knowledge, most models in semantics, including my own earlier work, do not make the distinction between content selection and generation. Even computational studies of generation (e.g. Jurafsky & Martin 2009: Chapter 24) seem to omit the content selection problem and simply start with a given content. Thus, while the sharp separation referred to above is an idealization, the key point is to recognize that *two* choices, not one, are involved, the choice of content and the choice of appropriate sentence to convey the content, whatever the particular manner in which the two are effected, whether separately as I have shown or together via an ordered pair in a single more complex game.

### 7 Defining Communication Games

## **7.4 The Interpretation Game**

Once A has publicly uttered , B presumably hears (or reads) it. Based on her using the same four Constraints – Phonetic, Syntactic, Semantic, and Flow – that the speaker has used, this leads to the public emergence of a game between A and B. There are, in fact, three games: one is the actual game () that comes into existence and the other two A (), B () are A's and B's models of their interaction. That is, the local games contained in the global games have three aspects and each of them derives from the game maps , A , and B . The "objective" map is not in either agent's private information but can be said to exist on account of the utterance. The construction of B () is very similar to the construction of A () above. Just as A's locutionary global game

$$LG\_u^{\mathcal{A}}(\varphi) = \{ \mathbf{g}\_1^{\mathcal{A}}, \mathbf{g}\_2^{\mathcal{A}}, \mathbf{g}\_{12}^{\mathcal{A}}, \mathbf{g}\_1'^{\mathcal{A}}, \mathbf{g}\_2'^{\mathcal{A}}, \mathbf{g}\_{12}'^{\mathcal{A}} \}$$

so B's locutionary global game is

$$LG^{\mathcal{B}}\_{\
u}(\boldsymbol{\varrho}) = \{ \mathbf{g}^{\mathcal{B}}\_{1}, \mathbf{g}^{\mathcal{B}}\_{2}, \mathbf{g}^{\mathcal{B}}\_{12}, \mathbf{g}'^{\mathcal{B}}\_{1}, \mathbf{g}'^{\mathcal{B}}\_{2}, \mathbf{g}'^{\mathcal{B}}\_{12} \}$$

and the objective locutionary global game is

$$LG\_{\mathfrak{u}}(\varphi) = \{\mathbf{g}\_1, \mathbf{g}\_2, \mathbf{g}\_{12}, \mathbf{g}\_1', \mathbf{g}\_2', \mathbf{g}\_{12}'\}.$$

B receives just this one sentence and does not need to consider other alternatives such as ′ or ″ as A did in (). So the Interpretation Game is just () = B (). 10 Thus, there is an asymmetry between A's and B's effort in communication: the speaker generally has to work harder. Incidentally, this slight time lag between A () being played in A's head and () appearing publicly and B () emerging in B's head after the utterance and the possibility of there not being common knowledge is a third reason why these games of partial information are not games of incomplete information (or signaling games).

The objective game () is made up of those parts of A's and B's subjective games A () and B () that are *known* to each of them. For example, in each game in A () there will be a part that involves A's own possible actions which would clearly be known to him. And there will be a part where he models B's possible actions which may or may not be accurately known to him. On the other hand, in each game in B () there will be a part that involves B's own possible actions which would clearly be known to her. And there will be a part where she

10I use the letter as a mnemonic for "understanding" as the letter has already been used for the illocutionary global game ().

### 7.4 The Interpretation Game

models A's possible actions which may or may not be accurately known to her. () will then consist of the parts of each interlocutor's subjective games that are known to them, A's own possible actions from A () and B's own possible actions from B (). This kind of setup for interpretation is more general than Grice's understanding, for example, as it makes possible a formal modeling of miscommunication and weaker flows of information in a natural way.

If all goes well, () = A () = B () and, indeed, they all become (nonconscious) common knowledge between A and B. So far, I have described just the locutionary global game A () in the previous section and said that the illocutionary global game A () will be developed in Part IV. But it is the global game () = () ∪ () that unfolds publicly and B () = B () ∪ B () that is induced privately in B. I will generally assume that the objective and subjective games are all identical and common knowledge as this is what happens when the communication succeeds. But there are situations where there is miscommunication or a weaker flow of information and this is then explained by the divergence among these three games or by the lack of common knowledge.

Prior to the utterance, A merely believed (nonconsciously) that the emergent game () would be common knowledge. When this belief turns out to be informationally caused,11 that is, when it is knowledge, then () will be common knowledge between A and B. This would mean that B's corresponding (nonconscious) belief that () is common knowledge would also be informationally caused. In this sense, common knowledge of the global game is not something that is exogenously assumed as is usually the case in applications of game theory but it is something that issues from the utterance of in the ambient utterance situation when communication succeeds, and fails to obtain when there is miscommunication or when there is a flow of information that is weaker than full communication. There could be a variety of factors responsible for miscommunication, either differences in the prior probability estimates each agent uses in each local game belonging to the global game or differences in the Phonetic, Syntactic, or Semantic Constraint outputs (e.g. the addressee might consider an additional word or parse or meaning or might fail to consider some possibility owing to differences in the agents' knowledge of the language) or other differences in their subjective games. Whatever the factor, common knowledge of the game will then not obtain. I have considered miscommunication in some detail earlier so I will not deal with it in this book.12 In the case of weaker information flows, the content does get through but it does not become common knowledge.

11Here "information" is used in Dretske's sense of being a true proposition. See Section 2.1.

12See Parikh (2001: Chapter 9) and Parikh (2010: Section 5.3).

### 7 Defining Communication Games

This happens when, for example, A suggests or hints something. I have also considered such weaker flows earlier though I will address them again briefly in Chapter 20. 13 Generally, I will just assume throughout that common knowledge does materialize.

Incidentally, common knowledge of () = A () = B () is not so hard to come by because the utterance itself is generally public and therefore common knowledge, the language is sufficiently shared so that the Syntactic and Semantic Constraints are (nonconscious) common knowledge, and therefore the possible contents considered in the various local games are common knowledge, and the payoffs are determined endogenously from A's possible intentions which are given by the possible contents, which leaves just the prior probabilities. These are specified by a host of subjective and objective factors and need to be just roughly shared. However, they are the one item that can go awry and so when there is miscommunication it is more often than not because the probabilities were differently assessed by the interlocutors. As I said in Section 2.2, there are sufficient grounds for saying that often enough there will be a situation ⊂ such that (a) ⊨ (), (b) ⊨ A knows , and (c) ⊨ B knows , which means that often enough A and B will have common knowledge of (), especially its locutionary part (), and, consequently, (partial) communication will succeed.

As before, I will postpone discussing the solution process to the Interpretation Game. Indeed, the solution process is a joint one between A and B and is carried out against the background of (), the objective global game. Given the assumed identity of the three global games, we can say for the moment that their joint solution is ℓ = 12 = *Bill Smith stood (for election)* and = 12 = 12 = [S[NP Bill][VP ran]] and the full meaning = ℓ ⊙ = *Bill Smith ran in the local election*. In other words, both A and B derive the same semantic and syntactic content from the utterance.

After solving (), B has understood the utterance. In a narrow sense of communication, this is all that is involved. But in a wider sense, she still has to respond to the content. When I defined communication in Section 5.3, it was the narrow sense that mattered as communication is not concerned with how the addressee responds to the utterance even though her possible responses in the interactive structure we have called the Content Selection Game influence the content and therefore the sentence A chooses to convey.

As I said in Parikh (1987b: 46), the interpretive act by B in the Interpretation Game is not observable by A. As a result, given just this much, A cannot tell if

13See Parikh (2001: Section 6.5) and Parikh (2010: Section 5.10).

### 7.5 The Content Selection Game again

the communication was successful. It is only B's response in the Content Selection Game, if it is an observable act, that allows A to infer how B might have interpreted his utterance. Otherwise, he would need to wait for some public action such as an utterance by B in an ongoing conversation of the kind we are modeling or some other kind of perceptible action to infer how she might have understood him. Even with such observations, there will generally remain some uncertainty about B's precise interpretation just as B will remain unsure about precisely what A was conveying because there are often fundamental indeterminacies in communication as discussed in detail in Parikh (2010: Chapter 5). Such indeterminacies arise from a variety of sources as I pointed out there, from possibly different utterance situations the agents carve out (and concomitant differences in the games induced) as well as differences in their knowledge of the language that includes the phenomenon of vagueness to be discussed in Chapter 11. In other words, the divergences between what was conveyed and understood are related to uncertainties each agent has about the content in the other agent's mind.

I have tried to tread a fine line in the preceding paragraphs. On the one hand, common knowledge is, often enough, not that hard to come by, but, on the other, it also does often fall short of what is required for an exact identity between what is conveyed and what is understood. More of the locutionary content will generally be identical as opposed to the illocutionary content but even with the former there are differences that arise, especially on account of vagueness.

## **7.5 The Content Selection Game again**

Based on the content that B infers from (), she constructs the Content Selection Game shown in Figure 7.1 in Section 7.2 for the first time. Until this point in time only A has played the game as a private matter. He has had to convert the optimal content into a public utterance of in in the Generation Game to convey to B. She then infers from the utterance in the Interpretation Game and so comes to construct the Content Selection Game. This is part of the circularity in communication. In general, as I remarked in Section 7.2, her model of the Content Selection Game may differ from A's and there will again be the same circumstance of an objective game and two subjective games. But, to simplify things, I assume common knowledge of a single game. In the trivial interaction at hand, B has just one action , acceptance, to choose from, so she simply carries out that action. Even here, this response is invisible to A and so he has to wait for some other public action by B to infer not just the meaning

### 7 Defining Communication Games

she derived from his utterance but also whether she accepted this meaning or not. Of course, in this trivial game, A knows that his content will be accepted because that is the only available response.

When describing the Interpretation Game in the previous section, I had pointed to an asymmetry between A and B. The speaker has to potentially consider multiple sentences in the Generation Game as indicated by Figure 7.8 but the addressee need consider just the single utterance selected by the speaker. This means the addressee never has to consider alternative *utterances* as is commonplace in Gricean pragmatics (e.g. Horn 1972; 1984, Levinson 1983; 2000). However, as will become clear in Part IV, she does have to consider alternative *contents* when the Content Selection Game presents more complex possibilities. The reason for this somewhat subtle difference is that while it is reasonable for addressees to reflect on *what* a speaker may be conveying, it is idle for them to concern themselves with *how* a speaker may be conveying it as many different ways of expressing the same content may exist. And, given the compositional nature of generation and interpretation, this can lead to an inefficient regress in processing, as I argue in Section 8.2. It is because much Gricean work is not fleshed out in as much detail as Equilibrium Semantics provides that it becomes possible to ignore this issue.

Once B has responded, the circle of communication is over and both agents return to the Setting Game where the whole enterprise began. All of this generally takes no more than a few seconds in thought and speech for A and comprehension and response for B although a writer may take more time to formulate his utterance and a reader may take more time to digest a sentence. All that has been accomplished here is that A has communicated to B and she has accepted his communication.

Below, I will look at two slightly more complex Content Selection Games where both agents have a real choice of action.

## **7.6 Back to the Setting Game**

Once they are back in the Setting Game where they are discussing the local election and perhaps politics generally, A's wish to elicit a certain sort of response from B has been more or less fulfilled. They are now ready to begin the next round of their conversation where B might, on the basis of what she has just heard and the setting they are in, form her own wish to elicit a corresponding response from A, leading to a new Content Selection Game. If we were interested in modeling dialogue, this is how it would have to be done, allowing for crossreferences to the shared history the interlocutors build up. This shared history,

### 7.6 Back to the Setting Game

and particularly the various objects the agents have referred to, become salient for future utterances in the dialogue via what is called an *information state* which maintains a record of the past contents conveyed. There are naturally three such states – one objective one and two subjective ones which may be more or less identical and common knowledge.

I have implicitly described a simpler kind of dialogue where each participant utters just one sentence at a time but there is no difficulty in extending such a picture to alternating *discourses* by each agent. It is possible to allow even more general settings where the agents might be performing other nonverbal actions such as gestures or interrupting the flow by a side exchange with some other party or by some other intervening acts. All such settings involve a sequence of utterance situations rather than a single one. Such sequences are called *discourse* or *dialogue* situations and are denoted by .

I have so far not discussed the entire content conveyed in a communication, just the infon and the parse tree , the latter being transmitted nonconsciously. What actually happens is that an utterance describes a situation *as* supporting or containing a content . That is, what is conveyed is the *proposition* = ( ⊨ ) where is called the described situation.14 The content is tied to some situation that is also communicated. For example, if A and B are talking about New York politics generally, then the situation of New York politics will be the described situation. If they are discussing just the local election, then the situation involving the local election will be the described situation. I have said above that the full content = ℓ ⊙ = *Bill Smith ran in the local election* and this implies that = *in the local election* is an illocutionary content of the utterance. This in turn suggests that the described situation is some larger situation, either covering New York politics () or even politics generally ( ′ ) or possibly New York generally ( ″). But I could equally have said that the described situation is one involving the local election (0) in which case there is no need for the illocutionary content as it would be redundant. In other words, the described situation is generally quite indeterminate and is seldom fully shared by the agents. Further, whether something is an illocutionary content and occurs to the right of the turnstile in the proposition conveyed or whether it is just a part of the described situation and so occurs to the left of the turnstile is often indeterminate. That is, the proposition conveyed could be any of ⊨ or ′ ⊨ or ″ ⊨ or 0 ⊨ ℓ or even other possibilities.

As I said in Section 2.1, it is agents who carve out situations and it is agents who determine the boundaries of what they are discussing. These boundaries are

14I have used as a mnemonic for the situation supporting a content as the letter is reserved for the discourse or dialogue situation.

### 7 Defining Communication Games

generally fuzzy and differently identified by the two agents. These differences are in fact quite productive as they lead the interlocutors to pursue different possibilities in what they contribute to the conversation as different contents suggest themselves to each agent. If one agent thinks they are discussing politics generally ( ′ ), he may shift the discussion to the presidential election; if the other agent thinks they are discussing New York generally ( ″), she may shift the discussion to the excesses of Wall Street. As such, the described situation also keeps changing through a conversation, getting larger or smaller or going off in a different direction altogether.

It is propositions that are true or false (or indeterminate) so that if the described situation had for some reason been a local election in Mumbai ( ′ 0 ) then the proposition ′ 0 ⊨ ℓ that Bill Smith ran in the local election in Mumbai would have been false. Both versions of the proposition ⊨ and 0 ⊨ ℓ may well be true. This raises the question how the described situation is even roughly determined. Where does it come from? It is partly inferred from and the larger dialogue situation , it partly issues from itself, and it is partly just creatively constructed by each agent. There is considerable latitude in where its boundaries are fixed and what matters to the communication is just whether it supports the content or not. I will say more about this when I look at how truth plays a role in meaning in Section 10.4.

For the time being, I just point out that the utterance and dialogue situation are different from the described situation although both are part of the context as I said in Section 2.2. This is a more general and, in my view, more accurate setup than the framework of a context set introduced by Stalnaker (1998) which plays both roles of utterance situation and described situation and consequently has fewer degrees of freedom. As discussed in some detail in Parikh (2010: 65–66), if is made up of a sequence of utterances each taking place in utterance situations 1, 2, …, then, just as 1 contributes to constructing 1, 1 in turn contributes to modifying 1 to establish the next utterance situation 2, which modifies 1 and generates the next described situation 2, and so on.

## **7.7 The Communication Game**

We have come full circle and we can now appreciate the explanation for A's utterance and B's response. First, there is the setting in which everything takes place. Part of this setting is the Setting Game which, in this case, is just a casual conversation between A and B about a local election. This induces the Content Selection Game for A, maybe because it is A's turn to contribute something to the dialogue. A solves this game and identifies the equilibrium content to

### 7.7 The Communication Game

convey to B in order to evoke some response in her, something as minimal as reception or acceptance in this example. After this selection, A plays the induced Generation Game and converts into possible sentences he could utter in to convey this content. One of these is . He evaluates by mentally building a model of the corresponding global game determined by the four Constraints, Phonetic, Syntactic, Semantic, and Flow, and computing its net value based on the global game's value and the cost of the utterance. After possibly evaluating other candidates as well, he chooses the optimal sentence and publicly utters it. Upon hearing the utterance, B builds her mental model of the global game called the Interpretation Game based on the same four Constraints and solves it. If the communication is to succeed, the three global games – the objective game and the two subjective models of it – have to be identical and common knowledge and B arrives at the content . Based on this, she plays the Content Selection Game and chooses her best response, just acceptance in this example. Finally, both return to the Setting Game for further rounds of the conversation.

Thus, A, in attending to the Setting Game, starts with the Content Selection Game and then plays the Generation Game. B starts with the Interpretation Game and then plays the Content Selection Game. This double set of choices that both agents have to make – selecting a content and then the utterance for the speaker and choosing an interpretation and then an action for the addressee – are quite reasonable once one realizes that one choice in each set of choices would have been required even if telepathy were possible. However, in practice, all this happens nonconsciously and seamlessly, possibly going back and forth multiple times among the various games as small chunks rather than whole utterances are processed in real time.

This is more or less the full structure of communication. There can be many variations of it. It can be seen to be qualitatively quite rich but each part of it is relatively simple. That is, its building blocks are uncomplicated and easily understood but the interactions among them make the overall system quite complex. This is exactly what we should expect intuitively and, as will be seen below, the solution to all these games is also quite straightforward and readily implementable either in human neural structures or in artificial agents such as robots.

We can now go back to the chart that was displayed in Section 3.1.

### Utterance Situation

Setting Game

### 7 Defining Communication Games

It should now be very clear how this cycle of communication works for just a single utterance. It is useful to assign symbols to all the games involved, so we denote the Setting Game by and the Content Selection Game by . The Generation Game already has the name () and the Interpretation Game has the name (). In general, there will be other contents A will contemplate in but it is just the optimal one that induces the corresponding Generation Game (). Likewise, there may be multiple candidate sentences A may consider in () but it is only the optimal utterance of that induces the Interpretation Game (). So we can leave these optimal arguments in there as part of the definition of the whole game called the Communication Game which has these four interlinked games as components. The Communication Game is then Γ = (, , (), ()). But the content and sentence can also be suppressed if desired to yield just Γ = (, , , ). Recall that the global games A () = A () ∪ A (), B () = B () ∪ B () are part of () and (), respectively. I have yet to describe the illocutionary game component of these global games which will be taken up in Part IV. There is also an objective global game () = () ∪ () that is induced by A's public utterance of which I have deliberately left out of the chart above and from the definition of Γ. It forms a kind of backdrop to the subjective games that the agents actually solve but what happens is determined by the objective game which is the actual game that gets played as the subjective games are just partial models of it. Of course, we will be largely concerned with situations where all three games are identical. If we wish, this missing objective game can be made explicit by defining Γ = (, , (), (), ()) = (, , , , ). Finally, each global game is made up of several local games that are either speaker games or addressee games or the corresponding objective games. These are called games of partial information and it is their joint solution that plays a crucial role in the process of communication.

At this stage, we are in a position to say that communication (in its wide sense) basically involves the playing of the Communication Game Γ in some situation . It is as simple or as complex as that, whichever way you wish to see it.

## **8 Solving Communication Games**

## **8.1 Solving Locutionary Global Games**

Assuming common knowledge of the objective locutionary global game () = {1, 2, 12, ′ 1 , ′ 2 , ′ 12} from Section 7.3 to keep things simple, I now look at how this game can be solved. When there is common knowledge, A () = B () = (), so solving the objective game automatically implies a solution to the subjective games. As I said in that section, is a solution function that maps games into their solutions and [()] = {(1), (2), (12), ( ′ 1 ), ( ′ 2 ), ( ′ 12)} = {1, 2, 12, 1, 2, 12}. The solution to the syntactic games can be trivially obtained in this example so I will focus on the semantic games.

Figure 8.1: Semantic lexical game 1

Recall the first local semantic game 1 = (1) shown again in Figure 8.1. In 1, A utters 1 = bill and B has to choose between 1 = *Bill Smith* or ′ 1 = *Bill Jones*. The prior 1 = (1 ∣ 2, 1, 2; ) where the 2 is a variable ranging across the possible corresponding meanings 2, ′ 2 , and the 1, 2 are variables standing for the possible corresponding parse trees 1, 2, respectively. Likewise, the other prior 1 ′ = ( ′ 1 ∣ 2, 1, 2; ). Thus, there are different *versions* of essentially the same game 1 when 2, 1, or 2 take on different values whenever this is possible. In addition to choosing utterances and interpretations, the agents also have to choose the particular values of the conditioning variables in the prior

### 8 Solving Communication Games

probabilities, that is, they have to choose between the two versions of the game. This is what makes all the local games interdependent.

In solving such games, the players have to identify all the possible strategies they could adopt and then choose the best one based on what the other player might do. A strategy is just a specification of what an agent would do at each of their information sets and of what version of the game they would choose. A has two information sets, one at 1 and the other at 1 ′, so he has to specify what he would do at both of these. However, he has just one choice of action, 1, in each set so there is no real choice. On the other hand, he can choose between the two versions of 1 essentially by choosing 2 = 2 or 2 = ′ 2 , as the other conditioning variables 1 and 2 take on just one value 1 and 2 respectively, and as the parameter is fixed. So his two strategies are (1, 1, 2) and (1, 1, ′ 2 ) where the 1 is repeated in each strategy because that is the action in both 1 and 1 ′. Since 1 is the only possible action, it can just be ignored, and we can simply say he has a choice between 2 and ′ 2 . B has just one information set, the elongated oval, but she has two choices, 1 and ′ 1 , in that set and she has the same two versions of the game to choose from so her strategies are (1, 2), (1, ′ 2 ), ( ′ 1 , 2), and ( ′ 1 , ′ 2 ). A strategy profile is a combination of a strategy of A's and a strategy of B's. All of their strategies listed above are called *pure* strategies; *mixed* strategies involve a probability distribution on pure strategies. The description of a strategy here is a generalization of the usual definition because of the need to include the choice of version or of the values of the conditioning variables. This is another reason why such games are called games of *partial* information.

One way to solve such a game is to look for strategy profiles that no player will want to deviate from unilaterally. Such an equilibrium strategy, called a Nash equilibrium as we saw in Section 2.2, 1 will implicitly involve each agent taking account of what the other agent might do. As I said in Section 5.1, the key idea in decision-making is that an agent should maximize his expected utility modulo what the other agent does. So we have to compare the expected utilities of different strategies for each agent and choose the ones that neither will want to deviate from unilaterally. Because 1 is part of the family of interdependent games () = {1, 2, 12, ′ 1 , ′ 2 , ′ 12} and because each agent will want to choose the version of 1 that the other chooses and that is compatible with the solutions to the other local games, there is an additional compatibility condition that is imposed on the solution that requires that both players play the same version

1 Strictly speaking, the right solution concept is not quite a Nash equilibrium but a Perfect Bayesian equilibrium but, in the simple structure of a single partial information game, this nuance can be safely ignored. See Parikh (2010: Section 3.3.5).

### 8.1 Solving Locutionary Global Games

of each game and that this version have conditioning variable values that agree with the choices made by B in the other games. For example, since there is a choice in 1 between two values of the conditioning variable 2, either 2 = 2 or 2 = ′ 2 , the value chosen in equilibrium by both players must match with the interpretation chosen in 2, where B has a choice between interpreting the utterance of 2 as 2 or ′ 2 as shown in Figure 7.4. And such a requirement must be satisfied in all six interdependent games. This means that while it is correct to see each player as choosing the optimal version of each local game by specifying the values of the conditioning variables in the prior probability distribution, each of them will optimally want this choice to agree with the interpretive choices made by B in the other local games. So there are fewer degrees of freedom than might initially appear to be the case.

## **8.1.1 The two versions of** 1

To be completely clear about the choice of version in each agent's strategy, I show the versions explicitly in Figures 8.2 and 8.3. The prior probability functions 1 and 1 ′ have been replaced by the actual probabilities with different combinations of the conditioning variables.

Figure 8.2: The first version of the semantic lexical game 1

Consider Figure 8.2 which shows the first version of 1. Assuming this first version is being played by both players as the compatibility condition requires, we investigate which strategy profile is best. This leaves just one real choice to be made: B has to decide between 1 and ′ 1 . If B opts for 1 then A's expected utility will be (1 ∣ 2, 1, 2; )A + ( ′ 1 ∣ 2, 1, 2; ) ′ A by tracing the paths followed by starting with 1 through to the relevant terminal node resulting from the branch labeled 1 and the payoff A received there and by starting with 1 ′ through to the relevant terminal node resulting from the branch labeled 1 and

### 8 Solving Communication Games

Figure 8.3: The second version of the semantic lexical game 1

the payoff ′ A received there. If B opts for ′ 1 then A's expected utility will be (1 ∣ 2, 1, 2; )A+( ′ 1 ∣ 2, 1, 2; ) ′ A. B receives the same expected utilities with A replaced by B in the above expressions.

As I said on page 101 in Section 7.3, (1 ∣ 2, 1, 2; ) > ( ′ 1 ∣ 2, 1, 2; ) owing to the compatibility between the conditioning variables and parameter and the conditioned variable in the first probability as opposed to the second. If, in addition, it is assumed that the unprimed payoffs are identical to the corresponding primed payoffs for both players, that is, if A = ′ A, A = ′ A, B = ′ B , and B = ′ B 2 because these payoffs are endogenously generated by the agents trying to coordinate with A's intentions to convey 1 and ′ 1 in 1 and 1 ′ and because there is no preference for any particular result in the ongoing conversation in , it follows that (1 ∣ 2, 1, 2; )B +( ′ 1 ∣ 2, 1, 2; ) ′ B > (1 ∣ 2, 1, 2; )B + ( ′ 1 ∣ 2, 1, 2; ) ′ B and so B's 1 strategy is better than her ′ 1 strategy.3 In fact, if B chooses 1 in the first version of 1 then *both* agents will receive a higher expected payoff than if she chooses ′ 1 . So, with respect to the first version, (1, 1) is a Nash equilibrium as neither player will want to deviate unilaterally from it, A because there is nothing to deviate to given his single choice of utterance 1 and B because ′ 1 yields a lower expected utility to her. Therefore, all this part of the solution process involves is comparing two prior probabilities. Indeed, as I suggested earlier, their values could well be 5/6 and 1/6 and it would just be

2Actually, all that is required is that A − A = ′ A − ′ A and B − B = ′ B − ′ B as can be seen from transposing terms in the inequality (1 ∣ 2, 1, 2; )A + ( ′ 1 ∣ 2, 1, 2; ) ′ A > (1 ∣ 2, 1, 2; )A + ( ′ 1 ∣ 2, 1, 2; ) ′ A and in the corresponding inequality for B. This essentially makes the game a coordination game of the kind considered in Section 2.2.

3As was discussed on page 99, the payoffs are always greater than the corresponding payoffs, primed and unprimed, for both players because they represent the correct interpretation with respect to what A is conveying.

a matter of seeing which one is larger. This computation is quite undemanding, either for the human brain or for an artificial agent.

Exactly the same reasoning shows that in the second version of 1 it is ′ 1 that gives the higher expected utility to B (and to A as well) because (1 ∣ ′ 2 , 1, 2; ) < ( ′ 1 ∣ ′ 2 , 1, 2; ) as was established earlier. So (1, ′ 1 ) will be a Nash equilibrium in this version as neither player will want to deviate unilaterally from it. In fact, I had suggested that these probabilities could be 1/4 and 3/4, respectively.

So, depending on which version is chosen by A and B, either 1 or ′ 1 will be the equilibrium choice for B (and 1 is the only possible choice for A). A more formal way of putting this is that the equilibrium of 1 will be either (1, 1, 2) or (1, ′ 1 , ′ 2 ) where the choice of the conditioning variable between 2 and ′ 2 has been made explicit. Of course, the compatibility condition mentioned above means that this latter choice will be fixed in solving 2. This makes the interdependence among the games crystal clear. I repeat that the trees 1 and 2 do not play any real role in this example as they are the only choices for the parses of 1 and 2. However, if we wish to make their role explicit as well, we can say that the choice of equilibrium in 1 is between (1, 1, 2, 1, 2) and (1, ′ 1 , ′ 2 , 1, 2). Since only the real choices need to be mentioned, it is possible to drop 1, 1 and 2 from these Nash equilibria and write just (1, 2) and ( ′ 1 , ′ 2 ). Here, the first component of the equilibrium, 1 or ′ 1 , is selected in 1 and the second component of the equilibrium, 2 or ′ 2 , is chosen in 2 rather than 1 owing to the compatibility condition.

As the reader might anticipate, a similar scenario obtains with the two versions of 2. In one version, 2 will be the best option for B based on the inequality (2 ∣ 1, 1, 2; ) > ( ′ 2 ∣ 1, 1, 2; ) and, in the other version, ′ 2 will be the best option based on the inequality (2 ∣ ′ 1 , 1, 2; ) < ( ′ 2 ∣ ′ 1 , 1, 2; ). This time, by similar reasoning, we can conclude that (2, 1) and ( ′ 2 , ′ 1 ) are the two Nash equilibria the players have to choose between. Here, the first component of the equilibrium, 2 or ′ 2 , is selected in 2 and the second component of the equilibrium, 1 or ′ 1 , is chosen in 1 rather than 2 owing to the compatibility condition again.  An alternative way of putting

### 8 Solving Communication Games

this is that the first versions of 1 and 2 are compatible with each other and the second versions of 1 and 2 are compatible with each other. In yet other words, we have two Nash equilibria in 1 and we have two Nash equilibria in 2 with the unprimed contents mutually implying each other and the primed contents mutually implying each other. How do we choose between these two sets of compatible pairs? One way to eliminate one pair of Nash equilibria is by bringing in the Pareto criterion and looking for what I have called Pareto-Nash equilibria in Section 2.2, something Spence (2001) also recommends in a different context.4 The Pareto criterion states that a Nash equilibrium is Pareto-efficient if it is at least as good as the other Nash equilibrium for both players and is strictly better for at least one player. For (1, 2) to be the unique Pareto-Nash equilibrium of 1, we require that:

4Another possibility is to consider what are called risk-dominant Nash equilibria but they yield the same result under the assumptions we have made and are less intuitive. See Harsanyi & Selten (1988) and Parikh (2010: Section 3.3.5). More recently, so-called *iterated best response* reasoning has been considered by Franke (2009) and Jäger (2012). While such reasoning can occur in games where agents consciously reason about one another, it is highly implausible in the context of communication which takes only milliseconds. Besides, as Jäger communicated to me, it is -complete. As we will soon see, while it is necessary to go through the Pareto-Nash calculations that I am describing, their final result is dramatically simple to compute and also does not involve each agent consciously reasoning about the other explicitly. In particular, while Pareto-Nash equilibrium is also -complete, in our context of communication, its complexity class is just .

Some theorists feel queasy about employing the Pareto criterion or even allowing agents to choose among Nash equilibria on the ground that agents can choose between strategies but cannot choose between strategy profiles, since the latter also include the other agent's choice. I have discussed this and related issues at some length in Parikh (2001: footnote 9, pages 40–41) and in Parikh (2010: Section 3.3.5) but here are some further observations. The central worry seems to be that some element of cooperation between the agents is required for the agents to choose Pareto-Nash equilibria in coordination games of the kind we are concerned with here. This may well be true. Earlier, I have suggested that such cooperation could result from one aspect of Grice's Cooperation Principle, which is being assumed in any case.

Agents can nevertheless sometimes choose different equilibria, and when they do, there is miscommunication. I discuss this kind of error in Section 8.4 in some detail. Often, it will lead to clarification requests, as Benz (2012) points out, and then the right equilibrium gets explicitly selected. It is unlikely, however, that both agents will get stuck jointly in a suboptimal equilibrium because the speaker knows what he intends to convey and his payoffs and probabilities will reflect this and make his choice optimal.

Given the alternatives – risk-dominance or epistemic reasoning of the iterated best response kind – assuming an element of cooperation in the choice of equilibria which, as we shall soon see, amounts to no more than choosing the maximal options, seems rather innocuous. Many game theorists such as Spence are not unduly troubled by this. I had mentioned in Parikh (2006a: 108–109) that the calculations in van Rooij (2004)showing the inadequacy of the Pareto criterion were flawed, as kindly conveyed to me by van Rooij himself.

either

$$\begin{aligned} &P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u)a\_{\mathcal{A}} + P(\sigma\_1' \mid \sigma\_2, t\_1, t\_2; u)c\_{\mathcal{A}}') \\ &> P(\sigma\_1 \mid \sigma\_2', t\_1, t\_2; u)c\_{\mathcal{A}} + P(\sigma\_1' \mid \sigma\_2', t\_1, t\_2; u)a\_{\mathcal{A}}') \end{aligned}$$

and

$$\begin{aligned} &P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u) a\_{\mathcal{B}} + P(\sigma'\_1 \mid \sigma\_2, t\_1, t\_2; u) c'\_{\mathcal{B}}) \\ &\simeq P(\sigma\_1 \mid \sigma'\_2, t\_1, t\_2; u) c \mathfrak{g} + P(\sigma'\_1 \mid \sigma'\_2, t\_1, t\_2; u) a'\_{\mathcal{B}}) \end{aligned}$$

or

$$\begin{aligned} &P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u)a\_{\mathcal{A}} + P(\sigma'\_1 \mid \sigma\_2, t\_1, t\_2; u)c'\_{\mathcal{A}}) \\ &\simeq P(\sigma\_1 \mid \sigma'\_2, t\_1, t\_2; u)c\_{\mathcal{A}} + P(\sigma'\_1 \mid \sigma'\_2, t\_1, t\_2; u)a'\_{\mathcal{A}}) \end{aligned}$$

and

$$\begin{aligned} &P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u)a\_{\mathcal{B}} + P(\sigma'\_1 \mid \sigma\_2, t\_1, t\_2; u)c'\_{\mathcal{B}}, \\ &> P(\sigma\_1 \mid \sigma'\_2, t\_1, t\_2; u)c\mathfrak{g} + P(\sigma'\_1 \mid \sigma'\_2, t\_1, t\_2; u)a'\_{\mathcal{B}}, \end{aligned}$$

Likewise, for (2, 1) to be a Pareto-Nash equilibrium of 2, we require that: either

$$\begin{aligned} &P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u)a\_{\mathcal{A}} + P(\sigma\_2' \mid \sigma\_1, t\_1, t\_2; u)c\_{\mathcal{A}}' \\ &> P(\sigma\_2 \mid \sigma\_1', t\_1, t\_2; u)c\_{\mathcal{A}} + P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u)a\_{\mathcal{A}}' \end{aligned}$$

and

$$\begin{aligned} &P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u) a \mathfrak{g} + P(\sigma'\_2 \mid \sigma\_1, t\_1, t\_2; u) c'\_{\mathfrak{B}}, \\ &\simeq P(\sigma\_2 \mid \sigma'\_1, t\_1, t\_2; u) c \mathfrak{g} + P(\sigma'\_2 \mid \sigma'\_1, t\_1, t\_2; u) a'\_{\mathfrak{B}}, \end{aligned}$$

or

$$\begin{aligned} &P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u)a\_{\mathcal{A}} + P(\sigma\_2' \mid \sigma\_1, t\_1, t\_2; u)c\_{\mathcal{A}}') \\ &\simeq P(\sigma\_2 \mid \sigma\_1', t\_1, t\_2; u)c\_{\mathcal{A}} + P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u)a\_{\mathcal{A}}') \end{aligned}$$

and

$$\begin{aligned} &P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u) a \mathfrak{g} + P(\sigma\_2' \mid \sigma\_1, t\_1, t\_2; u) c\_{\mathfrak{g}}' \\ &> P(\sigma\_2 \mid \sigma\_1', t\_1, t\_2; u) c \mathfrak{g} + P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u) a\_{\mathfrak{g}}' \end{aligned}$$

### 8 Solving Communication Games

There are so many symbols in these inequalities that it is hard to decipher what they amount to. It is convenient to let (1, 2 ∣1, 2; ) = , (1, ′ 2 ∣1, 2; ) = , ( ′ 1 , 2 ∣1, 2; ) = , and ( ′ 1 , ′ 2 ∣1, 2; ) = in order to see the probabilities more clearly. This implies:

$$\begin{aligned} P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u) &= \frac{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_2 \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)} \\ &= \frac{j}{j + l} \end{aligned}$$

$$\begin{aligned} P(\sigma\_1 \mid \sigma\_2', t\_1, t\_2; u) &= \frac{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_2' \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{k}{k+m} \end{aligned}$$

$$\begin{aligned} P(\sigma\_1' \mid \sigma\_2, t\_1, t\_2; u) &= \frac{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_2 \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)} \\ &= \frac{l}{j + l} \end{aligned}$$

$$\begin{split}P(\sigma\_1' \mid \sigma\_2', t\_1, t\_2; u) &= \frac{P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_2' \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{m}{k + m} \end{split}$$

$$\begin{aligned} P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u) &= \frac{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1 \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{j}{j + k} \end{aligned}$$

$$\begin{aligned} P(\sigma\_2 \mid \sigma\_1', t\_1, t\_2; u) &= \frac{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1' \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)}{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{l}{l+m} \end{aligned}$$

$$\begin{aligned} P(\sigma\_2' \mid \sigma\_1, t\_1, t\_2; u) &= \frac{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1 \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{k}{j + k} \end{aligned}$$

$$\begin{split} P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u) &= \frac{P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1' \mid t\_1, t\_2; u)} = \frac{P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)}{P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u) + P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)} \\ &= \frac{m}{l+m} \end{split}$$

Now we can substitute these simpler and more transparently interrelated fractions into the inequalities above. The first set of inequalities becomes: either

$$\frac{j}{j+l}a\_{\mathcal{A}} + \frac{l}{j+l}c'\_{\mathcal{A}} > \frac{k}{k+m}c\_{\mathcal{A}} + \frac{m}{k+m}a'\_{\mathcal{A}}$$

and

$$\frac{j}{j+l}a\_{\mathcal{B}} + \frac{l}{j+l}c'\_{\mathcal{B}} \simeq \frac{k}{k+m}c\_{\mathcal{B}} + \frac{m}{k+m}a'\_{\mathcal{B}}$$

or

$$\frac{j}{j+l}a\_{\mathcal{A}} + \frac{l}{j+l}c'\_{\mathcal{A}} \simeq \frac{k}{k+m}c\_{\mathcal{A}} + \frac{m}{k+m}a'\_{\mathcal{A}}$$

and

$$\frac{j}{j+l}a\_{\mathcal{B}} + \frac{l}{j+l}c'\_{\mathcal{B}} > \frac{k}{k+m}c\_{\mathcal{B}} + \frac{m}{k+m}a'\_{\mathcal{B}}$$

The second set of inequalities becomes: either

$$\frac{j}{j+k}a\_{\mathcal{A}} + \frac{k}{j+k}c'\_{\mathcal{A}} > \frac{l}{l+m}c\_{\mathcal{A}} + \frac{m}{l+m}a'\_{\mathcal{A}}$$

and

or

$$\frac{j}{j+k}a\mathfrak{g} + \frac{k}{j+k}c'\_{\mathfrak{B}} \simeq \frac{l}{l+m}c\mathfrak{g} + \frac{m}{l+m}a'\_{\mathfrak{B}}$$

$$\frac{j}{j+k}a\_{\mathcal{A}} + \frac{k}{j+k}c'\_{\mathcal{A}} \simeq \frac{l}{l+m}c\_{\mathcal{A}} + \frac{m}{l+m}a'\_{\mathcal{A}}$$

and

$$\frac{j}{j+k}ag\mathfrak{g} + \frac{k}{j+k}c'\_{\mathfrak{B}} > \frac{l}{l+m}cg\mathfrak{g} + \frac{m}{l+m}a'\_{\mathfrak{B}}\mathfrak{g}$$

### 8 Solving Communication Games

Recalling that A = ′ A > ′ A = A and B = ′ B > ′ B = B, cross-multiplying and simplifying, and combining the two sets of inequalities, we get:

$$\{jk > lm \text{ and } jk > km\} \text{ for } \mathcal{A} \text{ and } \{jk \succeq lm \text{ and } jk \succeq km\} \text{ for } \mathcal{B}$$

or

$$(jk \simeq lm \text{ and } jl \simeq km) \text{ for } \mathcal{A} \text{ and } (jk > lm \text{ and } jl > km) \text{ for } \mathcal{B}$$

This further simplifies to:

(5) > and > for both agents

Now Equation 5 can be simplified in two different ways. First, multiplying the two inequalities, we get:

$$j^2kl > m^2kl$$

which implies:

>

As = (1, 2 ∣1, 2; ) and = ( ′ 1 , ′ 2 ∣1, 2; ), this tells us that for (1, 2) to be a Pareto-Nash equilibrium of 1 and for (2, 1) to simultaneously be a Pareto-Nash equilibrium of 2, we must have:

$$P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) > P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)$$

As already noted above, (1 ∣ 2, 1, 2; ) > ( ′ 1 ∣ 2, 1, 2; ) and (1 ∣ ′ 2 , 1, 2; ) < ( ′ 1 ∣ ′ 2 , 1, 2; ). This implies:

$$P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) > P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u)$$

and

$$P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) > P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u)$$

So (1, 2 ∣1, 2; ) is the maximum among

$$\left\{ P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u) \right\}.$$

Looking back to Figure 7.6, this simply means that for (1, 2) to be a Pareto-Nash equilibrium of 1 and for (2, 1) to be a Pareto-Nash equilibrium for 2, all that is required is that the prior probability 12 = (1, 2 ∣1, 2; ) be the

8.1 Solving Locutionary Global Games

maximum among the four priors in the product game 12.This result could also be obtained by solving 12 directly for its Nash equilibrium which can be seen to be 12 by similar computations. Importantly, in examples with syntactic ambiguity we would need to consider the full mixed semantic-syntactic sentential product mentioned on page 110 and compute its Nash equilibrium.5

Keep in mind that this is the result when these pairs form the unique Pareto-Nash equilibria of their respective games. There is no guarantee that this will always happen. It is quite possible, in a particular locutionary game, to have (1, 2 ∣1, 2; ) = ( ′ 1 , ′ 2 ∣1, 2; ). In such cases, the best that can be said is that *both* sets of solutions are plausible contents of the communication. This can certainly happen, and does, for example, with puns. But, in the communication at hand, it can safely be asserted that:

$$P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u) > P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u)$$

and that (1, 2 ∣1, 2; ) is the *strict* maximum of

$$\left\{ P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u) \right\}$$

given the nature of . This is because the implicit shared goal of is to discuss the local election and so it gives a bigger boost to (1, 2 ∣1, 2; ) than to ( ′ 1 , ′ 2 ∣1, 2; ). In other words, when it is warranted by , the Pareto-Nash inequalities will naturally yield a strict or unique maximum, and when it is not warranted, they will not.

An alternative manipulation of Equation 5 implies:

$$jk + jm > jm + lm \quad \text{and} \quad jl + jm > jm + km$$

and so:

$$\frac{j}{j+l} > \frac{m}{k+m} \quad \text{and} \quad \frac{j}{j+k} > \frac{m}{l+m}.$$

which is just:

$$P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u) > P(\sigma\_1' \mid \sigma\_2', t\_1, t\_2; u) \text{ and } P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u) > P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u)$$

Therefore (1 ∣ 2, 1, 2; ) is the maximum among the four priors

$$\left\{ P(\sigma\_1 \mid \sigma\_2, t\_1, t\_2; u), P(\sigma\_1' \mid \sigma\_2, t\_1, t\_2; u), P(\sigma\_1 \mid \sigma\_2', t\_1, t\_2; u), P(\sigma\_1' \mid \sigma\_2', t\_1, t\_2; u) \right\}$$

5 See Chapter 9.

### 8 Solving Communication Games

in the two versions of 1 and (2 ∣ 1, 1, 2; ) is simultaneously the maximum among the four priors

$$\left\{ P(\sigma\_2 \mid \sigma\_1, t\_1, t\_2; u), P(\sigma\_2' \mid \sigma\_1, t\_1, t\_2; u), P(\sigma\_2 \mid \sigma\_1', t\_1, t\_2; u), P(\sigma\_2' \mid \sigma\_1', t\_1, t\_2; u) \right\}$$

in the two versions of 2.

## **8.1.3 A theorem**

Thus, () = {1, 2, 12, ′ 1 , ′ 2 , ′ 12} can be solved in two equivalent ways. We can confine our attention to just the lexical games and select the maximum priors across all versions in each game, ensuring that the priors are compatible, that is, that the conditioning variable in one is a conditioned variable in the other and vice versa. Or we can choose the maximum priors in the full mixed semanticsyntactic sentential product game. In the simple example at hand, the parse trees 1 and 2 did not play any real role so we were able to manage with just the semantic sentential product, but when they do, then the full mixed semantic-syntactic sentential product needs to be accessed. I record this important observation below.6

**Theorem 8.1.** The locutionary global game induced by an utterance can be solved either by solving just the lexical games in it or by solving just the full mixed sentential game or by solving appropriate intermediate phrasal games.

From both a psycholinguistic and computational viewpoint as well, all that has to be done is to work with either the conditional probabilities of the lexical games or the joint probabilities of the product games and select the corresponding maximum values. This gives us the solution to the locutionary global game in a completely rigorous and foundational way assuming nothing more than the rationality of agents.

The foregoing calculations show that there is a certain kind of overdetermination at work. First, the conditions for A's best actions yield the same results as the conditions for B's best actions so they are, in a sense, superfluous. And we know that A makes his choice of probability distributions or versions based on B's choices in the other games. However, this *is* a nontrivial decision and it is A who decides to let it depend on B's choices in the other local games. The results

6 Important statements are displayed as theorems rather than facts or propositions even though their proofs are not particularly complex. I do not actually give proofs of any of the theorems but I hope the example considered makes them plausible.

### 8.1 Solving Locutionary Global Games

therefore show that A's making his choices compatible with B's is in fact optimal from A's viewpoint. Incidentally, the same decision procedure in her choice of probability distribution or version works optimally for B as well.

## **8.1.4 The compact form**

The information contained in the locutionary global game () = {1, 2, 12, ′ 1 , ′ 2 , ′ 12} is scattered across six separate local games. But these games are highly interconnected and have a reciprocity through their prior probabilities. Is there a way to capture these dependencies in a more compact way? The answer is yes, as determined by the game theorist Ennio Stacchetti during a private discussion about (). He came up with a single representation that compactly marshals all the information dispersed among the six local games, assuming that A's and B's choice of version or probability distribution is compatible with B's choices in the other games. I call the former representation the *distributed* form of () and Stacchetti's representation the *compact* form of (). The existence of the compact form shows that it is right to think of () as a single game. I have tried to motivate this a little via the various calculations above. The compact form is shown in Figure 8.4. 7

Figure 8.4: The compact form of ()

In this new, more abstract representation of (), the players are no longer A and B but may be said to be the four lexical variables 1, 2, 1, 2 corresponding to the semantic and syntactic contents of the two words bill and ran in the sentence uttered in . (Alternatively, the words could be said to be the players but each word would be taken as having two avatars, one semantic and one syntactic.) Readers new to game theory may feel that the players in a game have to

7This representation can also be referred to as a normal or strategic form. It is in fact both, a compact form and a normal form, the first because of its relation to the distributed form and the second because it is itself related to a so-called extensive form which is different from the distributed form. In other words, the distributed form can be translated into a single extensive form as well. Thus, we have the distributed form ∼ the compact form ≡ the normal form ∼ the extensive form.

### 8 Solving Communication Games

be *agents* as ordinarily conceived but it is possible to allow games to have a more abstract kind of agency with more abstract kinds of players as we have here.

Recall that 1 stood for either 1 or ′ 1 and these are the two strategies available to the player 1. Similarly, 2 has the two strategies 2 and ′ 2 . These two sets of strategies are shown on the left and on top of the payoff matrix. The other two players 1 and 2 have just single strategies 1 and 2 so they are shown in the upper left corner of Figure 8.4 in order to avoid having to present a fourdimensional matrix with the third dimension labeled 1 and the fourth 2. Instead of a four-dimensional matrix, multiple two-dimensional matrices can also represent the strategies of these players were they to have some real choice. In general, for a sentence with words there would be a 2-dimensional matrix because there would be semantic variables and syntactic variables. The payoffs for the various combinations of strategies are shown in the cells of the matrix. For example, in the first cell, we have the vector ((1 ∣ 2, 1, 2; ), (2 ∣ 1, 1, 2; ), 1, 1) which indicates that the payoff to 1 is the first component, the payoff to 2 is the second component, and the payoffs to 1 and 2 are just 1 and 1 because the relevant conditional probabilities (1 ∣ 1, 2, 2; ) = 1 as shown in Figure 7.3 and (2 ∣ 1, 2, 1; ) = 1 as shown in Figure 7.5.

The somewhat complicated mass of symbols in Figure 8.4 can be simplified if we drop the players 1, 2 as they are just silent bystanders and make the substitution of letters for conditional probabilities I had suggested above. The result is shown in Figure 8.5.

Figure 8.5: The compact form of () with simpler symbols

This is now a familiar sort of game with a payoff matrix for two players. From the distribution of payoffs, it can be immediately seen that there must be at least one pure strategy Nash equilibrium. This is because a Nash equilibrium implies that neither player would want to deviate unilaterally from it. If no such Nash equilibrium existed, there would have to be a cycle with strategy profile (1, 2) dominated by ( ′ 1 , 2) for the first player (that is, the first player would deviate unilaterally from its strategy 1 to another strategy ′ 1 ), which in turn would be

dominated by ( ′ 1 , ′ 2 ) for the second player, which would itself be dominated by (1, ′ 2 ) for the first player again, which would finally be dominated by (1, 2) for the second player. In other words, there would be no "resting place" or Nash equilibrium. This would happen if the following inequalities among payoffs held, where the symbol "≺" for = 1, 2 stands for "dominated by for player ":

> (1, 2) ≺1 ( ′ 1 , 2) implies + < + ( ′ 1 , 2) ≺2 ( ′ 1 , ′ 2 ) implies + < + ( ′ 1 , ′ 2 ) ≺1 (1, ′ 2 ) implies + < + (1, ′ 2 ) ≺2 (1, 2) implies + < +

all of which together imply < < < < , a contradiction. This means there cannot be such a cycle. It should be clear by symmetry considerations that such an argument will apply to any compact form. I record this below.

**Theorem 8.2.** Every compact form of a locutionary global game has at least one Nash equilibrium in pure strategies.

For (1, 2) to be a Nash equilibrium, we must have:

$$\frac{j}{j+l} \simeq \frac{l}{j+l}$$

and

$$\frac{j}{j+k} \simeq \frac{k}{j+k}$$

These inequalities imply ≥ and ≥ . For ( ′ 1 , ′ 2 ) to be a Nash equilibrium, we must have:

$$\frac{m}{k+m} \ge \frac{k}{k+m}$$

$$\frac{m}{1+k} \ge \frac{l}{1+k}$$

and

 + + These inequalities imply ≥ and ≥ . If both (1, 2) and ( ′ 1 , ′ 2 ) are Nash equilibria – as we saw earlier – then in order for the former to be the unique Pareto-Nash equilibrium, we must have:

### 8 Solving Communication Games

either

and

or

$$\frac{j}{j+k} \ge \frac{m}{l+m}$$

$$\frac{j}{j+l} \ge \frac{m}{k+m}$$

 +

 + >

and

$$\frac{j}{j+k} > \frac{m}{l+m}$$

This implies either > and ≥ or ≥ and > , both of which imply > , exactly as we had when we considered the distributed form. But notice how much easier it was to obtain the result.

So far, I have considered just the four lexical variables 1, 2, 1, 2 derived from the words in the sentence as players but there are, in fact, potentially phrasal and sentential variables too. These are 12 = 1 ⊙ 2 and 12 = 1 ⋆ 2. The four possible strategies of 12 are 12, 1 ′ 2 , ′ 12, and ′ 1 ′ 2 and the single strategy of 12 is 12. But since these choices of the sentential variables are in fact determined by the choices of the four lexical players, these sentential players have no *real* choice beyond the lexical choices. And, indeed, this is reflected in the fact that the payoffs the semantic sentential variables receive from the four strategy choices – (1, 2 ∣1, 2; ) = , (1, ′ 2 ∣1, 2; ) = , ( ′ 1 , 2 ∣1, 2; ) = , and ( ′ 1 , ′ 2 ∣1, 2; ) = – are also ordered by the requirements we have displayed above for various equilibria. For example, as we just saw, if (1, 2) is a Pareto-Nash equilibrium of the compact form, then (1, 2 ∣1, 2; ) = is automatically the maximum among

$$\begin{aligned} \left\{ P(\sigma\_1, \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1, \sigma\_2' \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2 \mid t\_1, t\_2; u), P(\sigma\_1', \sigma\_2' \mid t\_1, t\_2; u) \right\} \\ &= \{ j, k, l, m \}. \end{aligned}$$

This leads to an insight that is parallel to Theorem 8.1 above.

**Theorem 8.3.** The compact form of a locutionary global game contains all the information required to determine its equilibrium. There is no need to consider nonlexical players.

Theorems 8.2 and 8.3 immediately imply the next observation.

8.1 Solving Locutionary Global Games

**Theorem 8.4.** Every locutionary global game has at least one Nash equilibrium in pure strategies.

This result does not imply that mixed strategies may never appear as equilibria. When there is more than one Nash equilibrium in pure strategies, there may not be a unique Pareto-Nash equilibrium in pure strategies, in which case a unique mixed strategy equilibrium may be optimal.

## **8.1.5 The main theorems**

In general, a sentence can be expressed as 1○2○…○ = 12 … where each is a word and there are words in the sentence. If is uttered in some utterance situation , then each word has a range of possible referential meanings given by the Semantic Constraint. The variable stands for these possible meanings of and all such together form the *meaning vector* = (1, 2, … , ). Likewise, the possible parse trees of each word can be collected in the *parse vector* = (1, 2, … , ). Here, the range of each variable is given by the Syntactic Constraint.

The meaning and parse vectors and together form the *content vector* = (, ) with 2 components. This vector ranges over all the possible lexical meanings and parses of uttered in . Note that = + . Also, − = (1, 2, … , −1 , +1 , … , 2). That is, is dropped from in − . Finally, ⋆ represents the equilibrium meaning and parse of all the lexical items.

We are now ready to state the central result of this section based on all the calculations above.

**Theorem 8.5.** Given an utterance and its locutionary global game (with symmetric payoffs), its lexical meanings and parses are given by the following equivalent characterizations:

$$\text{(6)}\qquad z^\star = \operatorname\*{argmax}\_{\mathbf{z}} P(z\_k \mid z\_{-k}; \boldsymbol{\mu}) = \operatorname\*{argmax}\_{\mathbf{z}} P(z; \boldsymbol{\mu}), \quad k = 1, \ldots, 2n$$

When there is more than one solution, each solution is given an equal probability.8 The meaning and parse of the whole utterance can be obtained by computing the products of the lexical meanings and parses.

Equation 6 is called the Fundamental Equation of Equilibrium Linguistics. It provides a constraint on the beliefs of speakers and addressees in communication. The term argmax ( ∣ − ; ) after the first equality sign and the term

8 By Theorem 8.4, a solution always exists.

### 8 Solving Communication Games

argmax (; ) after the second equality sign refer implicitly to the lexical games and to the full mixed semantic-syntactic sentential product game in the locutionary global game. Equation 6 improves upon the equation in Parikh (2010: Section 7.4) where the equivalence between the two ways of calculating ⋆ remained implicit. In principle, all a computing agent has to do is to run through the finite number of alternatives one by one until one or more vectors ⋆ is found to solve the system. It would be very easy to extend it to the Phonetic Constraint and to phonetic contents. At the start of this section, I mentioned the solution function which allows us to write [()] = {(1), (2), (12), ( ′ 1 ), ( ′ 2 ), ( ′ 12)} = {1, 2, 12, 1, 2, 12}. This function is an equivalent way of expressing ⋆ .

The Fundamental Equation treats semantic, syntactic, (and phonetic) contents in a completely homogeneous manner. This is what justifies calling them all *contents* of the utterance. It also shows how the pipeline view of meaning – first phonetics, then syntax, then semantics, and last pragmatics – is completely transcended and replaced by a circular system of simultaneous equations. As I will show in Section 10.2, this is one way in which Equilibrium Linguistics generalizes Frege's principle of compositionality because Frege assumed that semantics reflects syntax but not the other way around. In Equilibrium Linguistics, all three contents – semantics, syntax, and phonetics – reflect one another. I do not show this right away as it is better to first complete the analysis fully and then discuss the philosophical consequences that emerge.

I had assumed earlier that A = ′ A, A = ′ A, B = ′ B , and B = ′ B , making the games coordination games. This is what allowed us to essentially ignore the payoffs and this is why we got the pure probabilistic result above. However, in general, these payoffs may not be symmetric because different outcomes corresponding to correct or incorrect interpretations may be valued differently by each agent. It is not difficult to bring the payoffs back in by simply scaling the conditional probabilities by appropriate factors in accordance with the inequalities for Pareto-Nash equilibria on page 133 as shown in Figure 8.6.

Figure 8.6: The scaled compact form of ()

### 8.1 Solving Locutionary Global Games

I will not analyze this scaled form of the locutionary game but just state the corresponding equation that results. Figure 8.6 reflects B's payoffs rather than A's though a corresponding matrix exists for A even though he has no choice of utterance. Let B (, ′ ) be B's payoffs in the lexical game (or ′ (−) if > and the game is syntactic rather than semantic) and B(, ′ ) be her payoffs in the full mixed sentential game: these payoffs vary with different values of the vectors and ′ , both of which represent full independent content vectors for . This notation compactly captures all the different individual payoffs such as B, ′ B , B, and ′ B and need not be symmetric, even allowing for variations across the different lexical games.

**Theorem 8.6.** Given a locutionary global game, the lexical meanings and parses of an utterance are given by the following equivalent characterizations:

$$\begin{aligned} z^\star &= \underset{z\_{-k}, z\_k'}{\text{argmax}} \sum\_{z\_k} P(z\_k \mid z\_{-k}; u) V\_k^{\mathcal{B}}(z, z\_k')\\ &= \underset{z'}{\text{argmax}} \sum\_z P(z; u) V^{\mathcal{B}}(z, z'), \quad k = 1, \dots, 2n \end{aligned}$$

When there is more than one solution, each solution is given an equal probability. The meaning and parse of the whole utterance can be obtained by computing the products of the lexical meanings and parses.

Equation 7 may be called the Fundamental Equation of Equilibrium Linguistics with Payoffs. It expresses a more general result of our analysis and can also be put in matrix form. Generally Equation 6 suffices although in certain contexts as pointed out in Section 12.2 we have to resort to Equation 7.

This completes my discussion of how locutionary global games are solved. The description has taken a few pages but the results are strikingly simple and in accord with our intuitions. All that needs to be done is to find the interpretations with the highest probabilities in a *compatible* manner where by "compatible" I mean the property that argmax ( ∣ − ; ) = argmax (; ) as described by Equation 6. One aspect of this – identifying the *most likely* contents – is certainly common in both human and artificial contexts. But the second aspect – maximizing these probabilities in a *compatible* manner – is less evident. That communication involves semantic, syntactic, and phonetic contents mutually determining one another via such an intricate probabilistic structure is remarkable.

Just to be clear that not every probability distribution is compatible, Figure 8.7 shows a distribution (1, 2) with two variables 1, 2 and three values in the range of each variable that is not.

### 8 Solving Communication Games

Figure 8.7: An incompatible distribution

For example, 1 could range over the contents 1, ′ 1 , ″ 1 corresponding to the three rows and 2 could range over the contents 2, ′ 2 , ″ 2 corresponding to the three columns. For this distribution,

$$\underset{z}{\operatorname{argmax}} \, P(z) = (\sigma\_1, \sigma\_2) \text{ but } \underset{z}{\operatorname{argmax}} \, P(z\_k \mid z\_{-k}) = (\sigma\_1'', \sigma\_2''),$$

as can be checked by inspection. That is, the maximum of the joint distribution occurs at the first row and first column and the maximum of the conditional distributions occurs at the third row and third column. Such a distribution is ruled out by Theorem 8.5.

Why do compatible probability distributions arise in communication? Recall from page 100 in Section 7.3 that each prior probability is *not* the conditional probability of some content but is rather the conditional probability of a speaker's *conveying* that content.That is, these prior distributions govern the *intentions* and *actions* of rational agents, of what they would and would not choose to convey or interpret if they were conveying or interpreting certain other things as well. So compatibility describes the two-sided *logic* of rational intention and action. It is co-extensive with rationality itself in the context of communication. Earlier, I had mentioned that in many settings agents are not perfectly rational but because the structures involved in locutionary global games are relatively simple and because the payoffs are also assumed to be symmetric, there is little scope for limited agents to falter. Even when scaled compact forms as shown in Figure 8.6 are used to derive the more general Equation 7, there is no reason for rationality to fail.

Equally important, these probability distributions have nothing to do with language per se. The same property would hold in any communication involving images, gestures, and other symbol systems. It may even apply beyond communication to appropriately linked subsystems of a larger system, as I conjecture in Parikh (2010: Section 7.1.4).

### 8.1 Solving Locutionary Global Games

Theorem 8.5 provides a theoretical foundation for and generalization of various observations in both psycholinguistics and statistical natural language processing, as I discuss in Chapter 12. On the other hand, to the best of my knowledge, the philosophy of language and theoretical linguistics and artificial intelligence do not appear to have entertained anything close to such ideas about how locutionary meanings and other contents might be derived from first principles. They usually just say such meanings are "conventional" and turn their attention to deriving implicatures using some version of Grice's conversational maxims. The problem of rigorously deriving content is a *philosophical* and *theoretical* problem, not just some detail to be relegated to empirical investigation.

Where do all these probabilities come from? For human interpretation, they would come partly from the grammar and partly from the utterance situation and wider discourse situation and would be based on both objective and subjective factors.9 People are generally quite good at rough probability estimates (as in figuring whether it is likely to rain by looking at the sky) and all they need to determine here is whether certain probabilities are higher than others. No precise numerical estimates are required. For an artificial agent, these probabilities would come from the grammar and from suitable corpora.

To a certain degree, all these probabilities rely on background knowledge of one kind or another, some of which may be present in the encyclopedic knowledge associated with the conventional meaning of a word and some of which may be present in one or more knowledge or belief bases in the agent's head. Some linguists may feel that the explanation has just been pushed back to some unknown probabilities but the problem of determining such probabilities is common to all action, not just communicative action, as should be clear from the *very* wide applicability and, indeed, applications of decision and game theory. Rather than say, as Chomsky has chosen to, that this makes the entire realm of human choice and behavior a mystery, one has to approach it in the usual way of science, which is to first divide the problem into two parts and then to tackle each separately. In the case of communication, I have shown how the problem of deriving locutionary meaning can be reduced to some probabilistic computations. Now, as computer scientists frequently do in limited domains, distributed and interconnected knowledge bases can be set up and it would become very clear how

9 See Mante et al. (2013) for some recent work on how contextual information is processed by the brain involving a recurrent neural network model and experiments with monkeys. See also http://engineering.stanford.edu/news/stanford-researchers-surprised-findhow-neural-circuits-zero-specific-information-needed-decisi. These findings, which are still in their infancy, are relevant for all the games in the book as they all make use of contextually derived probabilities.

### 8 Solving Communication Games

such probabilities can emerge from such data. If an agent knows that black clouds make rain likely and he sees black clouds, it follows that the probability of rain is high. That is all. Indeed, it is not even necessary to build an actual knowledge or belief base in the case of human communication; it suffices merely to assume that it exists and that neuroscience will, in time, enable us to understand how it figures in decision-making. Building satisfactory conversational agents based on Equilibrium Semantics does require constructing and *scaling* the knowledge base sufficiently but I think it is possible to say that the problem of determining locutionary meaning has been solved. I touch upon this point briefly again in Section 12.2.

An alternative response to this issue of explaining how such parameters arise is to say that the complaint misunderstands the scientific problem. In physics, for example, the solution to the problem of projectile motion is divided into two parts: the relevant equation of motion and the initial or boundary conditions (e.g. the initial velocity). The scientific problem is determining the equation of motion. In different situations, the initial or boundary conditions will be determinable with more or less precision but no one thinks it is reasonable to say that the problem has just been pushed back to some unknown parameters.

I reproduce a paragraph from my dissertation, *Language and Strategic Inference* (1987b: 5):

We will argue for this thesis by developing a detailed account of one strategic inference in isolation. Any complete utterance, that is, any utterance that attempts to express a proposition, will involve many separate acts and strategic inferences. For example, part of a communication will typically involve a referential act, an act of referring to some object, and the communication of this reference to the addressee. Each bit of information communicated will require its own strategic inference(s). Thus, any complete utterance involves an entire system of simultaneous strategic inferences. These inferences have to be simultaneous in general because they codetermine each other in general. For example, an utterance of "Bill has the book" will require inferring the designata of each of the four words in the sentence, (not to mention its internal structure), in order to determine the proposition expressed. No word has any particular priority in this determination. That is, there may be interactions among the various strategic inferences. And the embedding circumstances play a vital role in each inference. Mathematically, this amounts to a system of simultaneous equations.

Earlier, I didn't know how to work out this idea. Now, several years later, I do.

8.2 My former partial information games

As the Interpretation Game () = B () = B ()∪B () = ()∪ () = () owing to our assumption of equality of the subjective and objective games and common knowledge of them, we have solved the locutionary part of the Interpretation Game the addressee faces. Its illocutionary part will be addressed in Part IV. The Generation Game is just a little more complicated as multiple sentences may need to be evaluated and their costs taken into account.

## **8.2 My former partial information games**

Readers familiar with my earlier books will have noticed that a game like the one in Figure 8.1 would ordinarily have been shown as in Figure 8.8.

Figure 8.8: Semantic lexical game 1

In this game, there are two more branches with alternative utterances ′ 1 at 1 and ″ 1 at 1 ′. These are both *unambiguous* with respect to the ambiguity in 1 = bill. So ′ 1 = bill smith and ″ 1 = bill jones represent one set of possibilities for these alternatives. As they are unambiguous, there is only one interpretation of each possible, either 1 or ′ 1 . As they are costlier but involve correct interpretations, the resulting payoffs A, B and ′ A, ′ B are somewhat lower than the corresponding payoffs and higher than the corresponding payoffs.

The games I have drawn in this and the previous chapter do not have such alternative utterances for two reasons. When deriving locutionary meaning, if we allow an alternative like ′ 1 = bill smith, then this alternative itself needs to be interpreted word by word, leading to a possible regress and further disambiguations, which is unrealistic. In Parikh (2010: 88 and 120–121), I suggested dealing

### 8 Solving Communication Games

with such alternatives *holophrastically* but this is unnecessary. Alternative *contents* are anyway considered by both agents in the Content Selection Game and the speaker *does* occasionally consider more than one alternative sentence (or subsentential expression) in the Generation Game so further alternatives *within* the partial information game are not required. Secondly, such costs are best considered *outside* the locutionary global game () but within the Generation Game . This makes it possible to include more sources of cost.

This dropping of alternative utterances may seem superficially like a small change but it results in *very* different solution processes for these new games of partial information, as readers familiar with the earlier solution processes will no doubt have realized. This is what makes possible the two parts of the Fundamental Equation 6 and all the theorems in Section 8.1. Earlier, such finer calculations of equilibria were not possible.

Another consequence of this truncated form of the game without the alternatives is that such alternative utterances no longer need to be part of the definition of speaker meaning. The subjective global game A () in Definition 5.1 in Section 5.3 contains just the sentence .

Others using game theory in this field (e.g. Benz et al. 2006), largely following the form of my earlier games of partial information above, have also used this idea of alternative utterances, apparently without realizing that these alternatives would themselves need to be evaluated using the same methods that are being applied to the main sentence, which could lead to an undesirable regress. To avoid this, they would once again have to follow my suggestion of locating alternative contents in the Content Selection Game and alternative sentences in the Generation Game outside the local games of partial information.

## **8.3 An interesting complication**

While the results of Section 8.1 are quite powerful, they depend on a condition that has been left implicit.10 Consider an utterance of the following sentence:

(8) The boy saw the girl with a telescope. ()

As is well known, the prepositional phrase with a telescope can attach either to the verb saw (i.e. to the verb phrase saw the girl) or to the noun girl and so is syntactically ambiguous. But there is no syntactic ambiguity at the level of lexical categories if they are expressed as in an ordinary CFG. The product of

10I did this deliberately to simplify the discussion as will become clear in this section.

### 8.3 An interesting complication

the optimal lexical parses obtained via Theorem 8.5 would then always yield *two* sentential parses even though the situation in which is uttered may suffice to disambiguate between them.

This incomplete disambiguation can be completed by adding sufficient detail to the lexical categories so that the preposition with has two distinct representations rather than just one. These representations may be described as *grammatical roles*: when the preposition attaches to the verb saw (or the verb phrase saw the girl), the role is that of an *instrument*, and when the preposition attaches to the noun girl, the role is that of *accompaniment*. Such roles and other refinements of lexical categories arise naturally in feature-based grammars of the kind described in Shieber (1986) and Steedman (2001).

Here are some examples of feature structures from Shieber (1986: Chapter 3):

$$\begin{aligned} & \begin{bmatrix} \text{cat:} & \text{NP} \end{bmatrix} \\ & \begin{bmatrix} \text{cat:} & \text{NP} \\ & \text{agreement:} & \text{singular} \\ & \begin{bmatrix} \text{number:} & \text{singular} \\ \text{person:} & \text{third} \end{bmatrix} \end{aligned} \end{aligned} $$

Here the first feature structure is just our plain vanilla category of a noun phrase. The second one shows the added feature of *agreement*. The value of this second feature is itself a feature structure with features *number* and *person* and corresponding values *singular* and *third*. This means the noun phrase is singular and third person and would have to combine with an appropriate category and feature structure to form a larger expression. The first feature structure above carries less information than the second about the noun phrase in question and is said to subsume the latter, leading to a natural lattice structure. The second, more informative feature structure also shows how feature structures can possess a nesting relationship.

In our example of the two roles for the preposition with, we could define the following two different feature structures for such words:

```
[role: instrument]
[role: accompaniment]
```
These different structures would differentiate syntactically between the two uses of the preposition at the *lexical* level as required.

### 8 Solving Communication Games

Feature structures can be and often are as finely articulated at the lexical level as required. This feature-based way of describing a context-free grammar then allows all syntactic ambiguities to be represented at the lexical level and so the "lexical computations" of Theorem 8.5 – ⋆ = argmax ( ∣ − ; ) – suffice to give us exactly one solution when warranted.

Even when the sentence is indefinitely long, as in A1 saw that A2 saw that … that A saw the girl with a telescope for any ∈ ℕ, the features can be recursively defined to allow an indefinite degree of differentiation at the lexical level for different attachments of the preposition with so that the lexical computations of Theorem 8.5 can still go through as required. In other words, all that is needed for the earlier results to hold is a sufficiently detailed feature-based representation of the grammar . This is the condition that was left implicit in Section 8.1.

What happens when is not feature-based (or, in other words, the feature structures are minimal) and there is a lack of differentiation at the lexical level for sentences like and similar sentences? In that case, the *compatibility* of the probability distribution as specified above will have to be generalized so that it applies to nonlexical levels as well.

Assume there is an utterance of a sentence as before. = {1, 2, … , , + 1, … , 2} is the set of semantic and syntactic indices of which stand for the *possible* lexical meanings and parses of uttered in . Let the index set ⊂ be a *variable* set that ranges over the indices of the possible meanings and parses of at a lexical, phrasal, and sentential level. For example, in a simple sentence such as "John loves Mary," ∈ {{1}, {2}, {3}, {2, 3}, {1, 2, 3}, {4}, {5}, {6}, {5, 6}, {4, 5, 6}} where the indices {1}, {2}, {3} correspond to the three semantic indices of the three words in the sentence, the set {2, 3} corresponds to the semantic indices of the phrase loves mary, the set {1, 2, 3} corresponds to the semantic indices of the whole sentence, and the indices {4}, {5}, {6} correspond to the three syntactic indices of the three words in the sentence, the set {5, 6} corresponds to the syntactic indices of the phrase loves mary, the set {4, 5, 6} corresponds to the syntactic indices of the whole sentence. Clearly, the range of variation of will be determined by each possible parse of and is thus determined by the grammar . If we write () for this range for the th parse out of possible parses (e.g. 1 () = {{1}, {2}, {3}, {2, 3}, {1, 2, 3}, {4}, {5}, {6}, {5, 6}, {4, 5, 6}} for the simple example where = 1 possible parse), we can then say ∈ () for the th of possible parses. That is, the index set is always a member of a particular range of variation () based on which parse is being considered.

Define = { ∣ ∈ } and define ′ to be its complement with respect to . Again, for the simple example above, if = {2, 3} then = {2, 3} and 8.3 An interesting complication

′ = {1, 4, 5, 6}. In this case, ( ∣ ′ ; ) = (2, 3 ∣ 1, 4, 5, 6; ). Likewise, if = {4, 5, 6} then = {4, 5, 6}, ′ = {1, 2, 3}, and ( ∣ ′ ; ) = (4, 5, 6 ∣ 1, 2, 3; ).

While the two paragraphs above are notationally cumbersome, the basic idea is very simple. Earlier, we found just the lexical level adequate when the grammar was feature-based. When it is not, we need to access the indices at a phrasal and sentential level as well so that the computations of Theorem 8.5 can be extended to nonlexical levels in a way that provides a generalized notion of compatibility. What the more general statement of Theorem 8.7 requires is to run through the relevant indices at phrasal and sentential levels as well and the indices that are relevant at such levels are the ones that are dictated by the possible parses being considered.

**Theorem 8.7.** Given an utterance and its locutionary global game (with symmetric payoffs), when the relevant grammar is not feature-based, its lexical meanings and parses are given by the following equivalent characterizations:

$$\text{(9)}\qquad \mathsf{z}^{\star} = \operatorname\*{argmax}\_{\mathsf{z}} P(\mathsf{z}\_{\mathsf{K}\_{\theta}} \mid \mathsf{z}\_{\mathsf{K}\_{\theta}^{\prime}}; \mathsf{u}) = \operatorname\*{argmax}\_{\mathsf{z}} P(\mathsf{z}; \mathsf{u}), \quad \mathsf{K}\_{\theta} \in K\_{\mathsf{G}}^{I}(\mathsf{\{\rho\}}), \; \mathsf{l} \in L.$$

When there is more than one solution, each solution is given an equal probability.11 The meaning and parse of the whole utterance can be obtained by appropriately computing the products of the lexical, phrasal, or sentential meanings and parses.

I urge the reader to mentally run through the more complex computations involved with this extended idea of compatibility. Notice that the term

$$\underset{z}{\text{argmax}} \, P(z\_{K\_{\emptyset}} \mid z\_{K'\_{\emptyset}}; u)$$

in Equation 9 does not involve any mixed semantic-syntactic products whereas the term argmax (; ) does so that there is still a reduction in computational effort by the equality of the two. If we call indefinitely articulated feature-based grammars complex and ordinary context-free grammars simple and if we call the lexical computations of Theorem 8.5 simple and the generalized computations of Theorem 8.7 complex, then there is a nice and intuitive inverse relationship between grammars and computations that follows from these two theorems. Let the grammar of a language be and the corresponding computations of optimal meanings and parses be .

11Again, by Theorem 8.4, a solution always exists.

### 8 Solving Communication Games

**Theorem 8.8.** and are inversely related in their complexity: when is simple, is complex, and when is complex, is simple.

For the rest of the book, I will assume we have a sufficiently articulated featurebased grammar and that the computations are therefore simple. In Chapter 9, I will attend to an example of syntactic ambiguity in detail, something I have not addressed so far but there all the ambiguities will be present at the lexical level itself so that issues of the kind raised by sentences like will not arise. For the present, we have to complete our discussion of our primary example = bill ran and I return to this below.

## **8.4 Solving Generation Games**

As we saw in Section 7.3 and Figure 7.8, the speaker may consider more than one sentence and therefore more than one locutionary global game. Each such game has a value A [()] = A + A. Likewise, each corresponding illocutionary game will have some value and it is the total value A [()] that is computed. Each global game () comes with a cost A () and so the net value to A, A () = A [()]− A (), is determined. Then, the sentence with the highest net value is chosen.

The cost A () is a crucial factor that controls what is actually uttered.12 Incidentally, considerations of cost are entirely absent in Lewis (1969) and also in so-called cheap talk games in economics (e.g. Crawford & Sobel 1982; Farrell 1987; 1993; Farrell & Rabin 1996; Kartik 2009; Blume & Board 2013) and this is just one reason why they are unsuitable for modeling linguistic communication.13 Cost

12I had first pointed this out in Parikh (1987b).

13The setup in Crawford and Sobel is that there is a sender and a receiver. The sender observes a state which is equally likely to be either 0 or 1 (although the original account involves a continuous state space). He then reports the state to the receiver either truthfully or not. The receiver observes the report and makes a decision which can be any number. The receiver prefers decisions that are closest to the value of the state and the sender prefers decisions that are closest to the state plus a bias > 0.

There are two ways to view such a setup. The shared language between sender and receiver either does not constrain interpretation at all or contains expressions that have only single conventional meanings but then allows any interpretation.

Consider the first option. Suppose the sender observes a 0 and sends a 0. The receiver can interpret this expression "0" as either 0 or 1. This means there is *no* shared conventional meaning that the expression has that constrains its possible interpretations the way the word "bank" limits its possible locutionary interpretations to either *financial institution* or *land alongside a river*. In other words, the expression "0" is implicitly taken to be conventionally meaningless. That is, the whole language (in this case, just "0" and "1") has no conventional meanings and

### 8.4 Solving Generation Games

is based on a variety of factors, mostly objective ones such as the length and complexity of the sentence and the effort required to mentally process it and to physically produce it but also subjective ones especially related to the consequences of making certain information explicit and leaving certain information implicit.

It is possible that the primary decision is between what to make explicit or leave implicit and objective factors such as length, complexity, and mental and physical effort play a relatively smaller role. Conversational style, habits of speech, special contexts of speech and writing (e.g. poetry and fiction and nonfiction), and other aesthetic aspects are one factor that affect this primary decision and different communities and individuals may follow different patterns in how much is made explicit. This is a matter for sociolinguists and psycholinguists to dig deeper into.