#  Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game

Martin B. Haugh
Department of Analytics, Marketing & Operations
Imperial College Business School, Imperial College
<EMAIL_ADDRESS>Chun Wang
Department of Management Science and Engineering
School of Economics and Management, Tsinghua University
<EMAIL_ADDRESS>
Abstract

The game of darts has enjoyed great growth over the past decade with the
perception of darts moving from that of a pub game to a game that is regularly
scheduled on prime-time television in many countries including the U.K.,
Germany, the Netherlands, and Australia among others. The game of darts
involves strategic interactions between two players but to date the literature
has ignored these interactions. In this paper, we formulate and solve the game
of darts as a dynamic zero-sum-game (ZSG), and to the best of our knowledge we
are the first to do so. We also estimate individual skill models using a novel
data-set based on darts matches that were played by the top 16 professional
players in the world during the 2019 season. Using the fitted skill models and
our ZSG problem formulation, we quantify the importance of playing
strategically, i.e. taking into account the score and strategy of one’s
opponent, when computing an optimal strategy. For top professionals we find
that playing strategically results in an increase in win-probability of just
0.2% - 0.6% over a single leg but as much as 2.3% over a best-of-35 legs
match.

##  1 Introduction

In recent years the game of darts has experienced a surge in popularity with
the game now having a substantial presence in many countries including the
U.K., Ireland, Germany, the Netherlands, and Australia among others. Indeed a
recent headline in The Economist (2020) explains “How darts flew from pastime
to prime time” and how it is not uncommon today to have tens of thousands of
fans attend darts tournaments. The game is also becoming increasingly
attractive to women as evidenced by the exploits of Fallon Sherrock who in
2019 became the first woman to win a match at the PDC World Darts Championship
and who made it to the quarter-final of the Grand Slam of Darts in 2021. The
aforementioned article from The Economist also notes how in recent years darts
has become the second-most-watched sport over the Christmas period on Sky
Sports in the U.K., coming second only to soccer. (Sky Sports is the premium
sports television channel in the U.K. and is comparable to ESPN in the U.S.)
Further growth is also anticipated with professional tournaments now taking
place in Shanghai and scheduled to take place in Madison Square Garden in New
York in 2022.

At its core, darts is a game between two players who both start on a fixed
score (typically 501) and take it in turns to throw three darts at a dartboard
approximately 8 feet away. Loosely speaking, each dart scores a number of
points depending on where it lands on the board and these points are then
subtracted from the player’s score to yield an updated score. The first player
to have an updated score of exactly zero wins the game or leg but the winning
dart must be a so-called double. Darts is therefore a dynamic zero-sum game
(ZSG) game where two players race each other to a score of zero. This leads to
two approaches to modeling the game of darts:

  1. 1.

Formulate a Markov Decision Process (MDP) problem where the goal of a player
is simply to minimize the expected number of turns it takes to reach zero.
This is sub-optimal since the optimal policy for the MDP takes no account of
the score of his/her (hereafter “his”) opponent. Indeed it is well-known that
top players do take account of their opponent’s score when they are playing.

  2. 2.

Formulate the game of darts as a dynamic ZSG and solve for the equilibrium
strategies of both players. Alternatively, if we believe our opponent will
play some fixed policy, then we would like to solve for the best-response (BR)
policy against his fixed policy.

Because the game of darts involves strategic interactions between the two
players, it should be modeled as a dynamic ZSG, but to date the literature has
ignored these interactions and only considered MDP formulations. The central
contribution of our work is therefore to formulate and solve the game as a
ZSG.

In order to distinguish between the MDP and ZSG approaches, we will use the
term “strategic” when referring to the BR/ZSG policies, and will generally use
the term “non-strategic” (NS) when referring to optimal MDP policies that do
not account in any way for the opponent’s score or strategy. (This is a slight
abuse of notation as MDP policies are strategic in the sense that they do take
account of a player’s own score.)

Solving for BR and equilibrium policies is considerably more challenging than
simply taking the first approach and modeling the problem as an MDP.
Therefore, one of the central goals of this paper is to formulate and solve
the BR and ZSG problems for a darts leg, i.e. a single race to zero. A further
goal is to use the resulting ZSG solution to quantify the extent of sub-
optimality in following the MDP/NS approach. We suspect the extent of this
sub-optimality to be quite small on any given leg, since it is only when the
players get relatively close to zero that it may be necessary to account for
an opponent’s score when determining the optimal throwing policy. Nonetheless,
no one to date has quantified the extent of this sub-optimality. Moreover, it
is possible that a small sub-optimality gap on a single leg can translate to a
more considerable gap across a full darts match which typically consists of
many legs.

In order to tackle these goals we borrow and extend the skill modeling
approach of Tibshirani et al. (2011) (hereafter TPT). We use a novel data-set
based on the dart-throws of the top 16 professional players from the 2019
season to fit a simple extension of TPT’s bivariate normal skill model. With
estimated skill models for each of the 16 players, we then formulate and solve
the game of darts as a dynamic ZSG. Using the fitted skill models, we
construct NS, BR, and ZSG equilibrium strategies for each of the players. We
quantify the difference in win-probability between playing the ZSG equilibrium
strategy instead of the NS strategy against a player playing the ZSG
equilibrium strategy. Over a single leg this difference is quite small and on
the order of approximately 0.2% - 0.6% for the top professional players.
However, a darts match is typically played across many legs, and then this
0.2% - 0.6% difference across a single leg can translate to a difference of as
much as 2.3% across a best-of-35 legs match which is typical in many big
tournaments. We are therefore the first to quantify the strategic element,
i.e. the importance of taking an opponent’s score into account, when playing
the games of darts.

To date, there has been relatively little analytic work on the game of darts
in the literature although this has begun to change in recent years. The
earliest paper that we are aware of is the work by Kohler (1982) who uses an
MDP formulation and a branch-and-bound approach to solve a slightly simplified
version of the game. More recently Baird (2020) considered an MDP formulation
where the goal is to minimize the expected number of turns until checking out,
i.e. reaching a score of zero with the last throw being a double. Both Kohler
(1982) and Baird (2020) therefore ignore their opponent’s score when
constructing a policy and produce sub-optimal policies for the ZSG.

Stern and Wilcox (1997) and Percy (1999) considered the easier problem of
where on the dartboard to aim in order to maximize the score of an individual
dart. The former used the Weibull distribution to model and fit dart throws.
More recently TPT proposed several models based on the Gaussian and skew
Gaussian bivariate distributions to model the throwing skills of players. They
use the EM algorithm together with importance sampling to fit these
distributions to dart-throwing data.

Another recent development is the work by Ötting et al. (2020) who find some
evidence for a weak-form of the so-called “hot-hand” phenomenon in darts. In
particular, they find strong evidence for it during the three dart throws
within a turn, but they find little evidence for it persisting across turns.
(In Section 2 we clarify what we mean by a “turn”.)

More generally the OR/MS community has taken a great interest in data
analytics over the past couple of decades and concurrent with this has been a
great interest in sports analytics; see for example Wright (2009) and more
recently Cokins and Schrader (2017). Some specific examples of work in the
field include Kaplan and Garstka (2001) on office pools motivated by the
“March madness” NCAA basketball tournament, Kaplan et al. (2014) on ice
hockey, Chan and Singal (2016) on handicapping tennis, Che and Hendershott
(2008) on assigning possession of the football in the NFL, and more recently
Hunter et al. (2019), Bergman and Imbrogno (2017) and Haugh and Singal (2020)
on fantasy sports. We see this paper as adding to this growing literature on
OR/MS applications in sports.

Finally, this paper also relates to the substantial literature on dynamic
ZSGs. ZSGs occur in many contexts including interdiction games, e.g. Fischetti
et al. (2019) and Han et al. (2016), the management of communication networks,
e.g. Brumbaugh-Smith and Shier (2002), and the modeling of heads-up poker
which is one of the most important test arenas for the development of
artificial intelligence (AI) research. Great progress has been made in recent
years and in fact AI-based poker players are now capable of beating the best
professionals in the world; see for example Bowling et al. (2015) and Sandholm
(2015).

The remainder of this paper is organized as follows. In Section 2 we explain
the rules of darts while in Section 3 we describe our conditional Gaussian
skill-model for throwing darts. In Section 4 we formulate the ZSG problem and
then in Section 5 we describe our algorithm for solving it. We present our
numerical results in Section 6 and we conclude in Section 7 where we also
outline some directions for future research. Various technical details as well
as additional numerical results are provided in the appendices and online
appendices.

##  2 The Rules of Darts

In order to understand the game of darts, we need to understand how a single
dart scores. Figure 1 displays a standard dartboard to aid our discussion. The
score obtained by a dart depends entirely on where on the board the dart
lands. In particular, the dart-thrower does not declare a target although he
will invariably have an (undeclared) intended target. The small concentric
circles in the middle of the board define the “double bulls-eye” (DB) and the
“single bulls-eye” (SB) regions. If a dart lands in the DB region (the small
red circle) then it scores 50 points while a dart landing in the SB region
(the green annulus surrounding the DB region) scores 25.

Figure 1: A standard dartboard

Beyond the SB region, the dartboard is divided into 20 segments of equal area
corresponding to the numbers 1 to 20. If a dart lands in the “20” segment, for
example, then it will land in the treble twenty (T20) region, the double
twenty region (D20), or the single twenty region (S20) for scores of 60, 40,
or 20, respectively. The double region is the region between the two outermost
circles on the dartboard whilst the treble region is the region between the
two circles beyond the SB region. The single region is then the union of the
two disjoint regions between the SB and treble regions and between the treble
and double regions. If a dart lands beyond the double region or simply fails
to land on the board then it scores zero. (The distances from the center to
the various regions on the dartboard are provided in Appendix A.)

The rules of darts that we describe here are for the most commonly played form
of the game, namely “501”. This is the form of the game typically played in
pubs and most professional tournaments. In 501 each player starts on a score
of 501 and takes turns in throwing three darts. These three darts constitute a
“turn”. After a player’s turn, his scores on the three darts are added
together and subtracted from his score at the beginning of the turn. The first
player to reach a score of exactly zero wins the game as long as his final
dart is a double, i.e. a dart that lands on D1, D2, …, D20, or DB in which
case we say the player has checked out. If a player’s turn would result in an
updated score of either 1, 0 (but not achieved via a double on his last dart)
or a negative score, then we say the player has gone bust. In this case the
turn is invalidated, the player remains on the score he had prior to the turn
and his opponent then takes his turn. Note that it is possible for a player to
win without having to throw all three darts in his turn. For example, suppose
a player has a score of 20 just prior to his turn. If he then scores D10 with
the first dart of his turn he wins. Alternatively, if he missed D10 with his
first dart and scored S10, then he could still win with the second dart of his
turn by throwing a D5.

The game we have just described is known as a leg. In practice, darts matches
are typically played over many legs with the winner being the first to win
some fixed number of legs. Alternatively, some tournaments have a legs and
“sets” structure whereby the winner of the match is the first to win a fixed
number of sets and the winner of a set is the first to win a fixed number of
legs. This latter structure then is similar to other sports such as tennis.
Finally, because the player that throws first has an advantage, players
alternate in starting legs (and sets). For example, if player A begins the
first leg of the first set, then player B will begin the first leg of the
second set, etc. Similarly within a set, players alternate in starting each
leg. Regardless of the match structure, however, the key component of a darts
match is the leg and it should be clear that a player will maximize his
chances of winning a match if he optimizes his strategy for playing each leg.

##  3 A Skill Model for Dart Throwing

Taking the center of the DB region to be the origin, we will use ${\bf
a}\in\mathbb{R}^{2}$ to denote the target of the dart throw and
$(x,y)\in\mathbb{R}^{2}$ to denote the outcome of the throw. We then let
$p(x,y;\,{\bf a})$ denote the probability of the outcome $(x,y)$ given the
target ${\bf a}$. Given $p(x,y;\,{\bf a})$ it is then easy to compute the
distribution $p(z;\,{\bf a})$ where $z=g(x,y)$ maps $(x,y)$ to the dart score
$z$, e.g. D16, SB, T20, S7, etc. We do this by numerically integrating
$p(x,y;\,{\bf a})$ over the region of the dartboard defined by $z$. For
example, to compute $p(\mbox{T20};\,{\bf a})$ we integrate $p(x,y;\,{\bf a})$
over the T20 region of the dartboard. We will also let $h(z)\in\mathbb{N}$
denote the actual numerical score achieved by the dart. For example,
$h(\text{D16})=32$, $h(\text{T20})=60$, $h(\text{SB})=25$, $h(\text{S7})=7$,
etc. Given $p(z;\,{\bf a})$ it is also easy to compute the distribution
$p(h(z);\,{\bf a})$. Our goal then is to construct a model for $p(x,y;\,{\bf
a})$ that can also be easily estimated with available data.

TPT proposed several models for capturing the skill of a darts player. Their
first model assumed the outcome $(x,y)$ of the throw follows a bivariate
Gaussian distribution with mean ${\bf a}$ (the intended target) and covariance
matrix $\sigma^{2}{\bf I}$ where ${\bf I}$ is the $2\times 2$ identity matrix.
They also proposed the more general model

$\displaystyle(x,y)$ $\displaystyle\sim$ $\displaystyle\mbox{N}_{2}({\bf
a},{{\bm{\Sigma}}})$ (1)

where ${{\bm{\Sigma}}}\in\mathbb{R}^{2,2}$ is an arbitrary covariance matrix.
TPT proposed the use of the EM algorithm to estimate ${{\bm{\Sigma}}}$ in (1).
They assumed the intended target ${\bf a}$ was known for each data-point but
that only the result $z=g(x,y)$ was observed rather than the realized location
$(x,y)$. That only the $z$’s were observed is a very reasonable assumption
given the difficulty of measuring the $(x,y)$-coordinates of dart throws.
Indeed this is certainly true for the data-set we will use in our numerical
study in Section 6.

A simple model such as (1) will not be sufficiently rich for modeling the
skills of professional darts players, because these players tend to focus on
(and practice throwing at) specific parts of the darts board, e.g. T20, T19,
DB, etc. This means that their skill levels, as determined by
${{\bm{\Sigma}}}$, are likely to depend on ${\bf a}$, which is indeed what we
observe in the data-set that we use in Section 6. We therefore assume there
exists a finite number of regions of the board, say $R_{1},\ldots,R_{M}$, so
that (1) can then be generalized to

$\displaystyle(x,y)$ $\displaystyle\sim$ $\displaystyle\mbox{N}_{2}({\bf
a},{{\bm{\Sigma}}}_{m}),\ \ \mbox{ for }{\bf a}\in R_{m}.$ (2)

Note that (2) is as easy to estimate as (1) since we can partition our data-
set by regions $R_{m}$ with each region consisting of one or more target
regions.

Once we have fitted a model $p(x,y;{\bf a})$ such as (2) we can then easily
compute $p(z;{\bf a})$, the corresponding distribution over scores, and
$p(h(z);{\bf a})$, the corresponding distribution over numerical scores.

##  4 The Zero-Sum Game Formulation

Ultimately a darts match (or specifically a leg of a darts match) is a dynamic
ZSG where the two players are in a race to zero. This means that, at least in
principle, a player needs to account for his opponent’s strategy when choosing
his own strategy. For example, it is well known that a player’s decision to
target the DB will depend in part on his opponent’s score.

Before proceeding, we clarify the zero-sum nature of a darts leg. In practice,
the winner of the leg scores one point and the loser scores zero and clearly
these two numbers do not sum to zero. However, we could scale both prizes by a
factor of two and then subtract one from each of them to obtain prizes of
$(1,-1)$ which are now in zero-sum format. It should be clear that this
transformation alters nothing in the players’ objectives and so a leg of darts
can indeed be modelled as a zero-sum game. We will express the game objective
from the perspective of player A who wishes to maximize the probability that
he wins, i.e. that he reaches zero before player B. Player B therefore wishes
to minimize this probability.

These win-probabilities will be a function of the state ${\bf
s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},t,i,u)\in{\cal S}_{\mbox{\tiny
ZSG}}$, where $s_{\mbox{\tiny A}}$ and $s_{\mbox{\tiny B}}$ are the scores at
the beginning of the turn for A and B, respectively,
$t\in\\{\mbox{A},\mbox{B}\\}$ denotes whose turn it is, $i\in\\{1,2,3\\}$
denotes how many throws are left in player $t$’s turn, and $u$ is player $t$’s
cumulative score thus far within the current turn. The set ${\cal
S}_{\mbox{\tiny ZSG}}$ then denotes the state-space for the ZSG. Because we
are taking player A’s perspective, the rewards for the ZSG satisfy
$r(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},t,i,u)=1_{\\{s_{\mbox{\tiny
A}}=0\\}}$, where $1_{\\{\\}}$ is the indicator function. The states
$\\{s_{\mbox{\tiny A}}=0\\}$ and $\\{s_{\mbox{\tiny B}}=0\\}$ correspond to A
winning and B winning, respectively. Technically, we do not want to have
$\\{s_{\mbox{\tiny A}}=0\\}$ as an absorbing state as this would imply A
receives a reward of 1 for every period in perpetuity. Instead we assume that
after reaching the states $\\{s_{\mbox{\tiny A}}=0\\}$ or $\\{s_{\mbox{\tiny
B}}=0\\}$ we then immediately move to an absorbing state $\tau$ that is
rewardless.

We define our action-space ${\cal A}$ to be the set of possible targets on the
dartboard. In our setting, ${\cal A}$ is state-independent and we assume it is
finite, reflecting the fact that any human will only be able to reasonably
distinguish a finite number of targets on the dartboard. An admissible policy
$\pi:=\\{\pi_{1},\pi_{2},\ldots\\}$ for player $j\in\\{\mbox{A},\mbox{B}\\}$
is a policy where each $\pi_{k}$ is a decision rule defining the target of the
$k^{th}$ dart. That is, $\pi_{k}({\bf s})\in{\cal A}$ for any ${\bf s}\in{\cal
S}_{\mbox{\tiny ZSG}}$ with $t=j$ and $k\in\mathbb{N}^{+}$ denoting the
$k^{th}$ dart thrown in the leg. Recall that $t$ is the component of ${\bf s}$
denoting whose turn it is. If $t\not=j$, then it is the turn of $j$’s opponent
and we set $\pi_{k}({\bf s}):=\emptyset$ to reflect the fact that he does not
throw a dart in this state. Stationary strategies $\pi$ satisfy
$\pi_{1}=\pi_{2}=\cdots$ and in this case we will use $\pi$ to denote both the
(stationary) strategy as well as the decision rule $\pi({\bf s})$. We let
$\Pi$ denote the space of all admissible policies $\pi$.

Let $J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}}}({\bf s})$ be the probability
that player A wins the leg when A and B play strategies $\pi_{\mbox{\tiny A}}$
and $\pi_{\mbox{\tiny B}}$, respectively, and where ${\bf s}=(s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}},t,i,u)$ denotes the current state of the game. That is,
we define $J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}}}({\bf
s}):=\lim_{K\to\infty}\text{E}\left[\sum_{k=1}^{K}r({\bf s}_{k})\mid{\bf
s}_{1}={\bf s}\right]$, where E indicates expectation and $K$ denotes the
total number of permitted dart throws in a finite-horizon version of the
problem. The min-max and max-min values of the game are then defined to be

$\displaystyle\underline{J}({\bf s})$ $\displaystyle:=$
$\displaystyle\min_{\pi_{\mbox{\tiny B}}\in\Pi}\max_{\pi_{\mbox{\tiny
A}}\in\Pi}J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}}}({\bf s})$
$\displaystyle\overline{J}({\bf s})$ $\displaystyle:=$
$\displaystyle\max_{\pi_{\mbox{\tiny A}}\in\Pi}\min_{\pi_{\mbox{\tiny
B}}\in\Pi}J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}}}({\bf s}).$

The study of dynamic ZSG’s was initiated in a classic paper by Shapley (1953).
For infinite horizon discounted problems he showed $\underline{J}({\bf
s})=\overline{J}({\bf s})=:J^{*}({\bf s})$ and that there exist stationary
strategies $\pi_{\mbox{\tiny A}}^{*}$ and $\pi_{\mbox{\tiny B}}^{*}$
satisfying

$J^{{\tiny\pi_{\mbox{A}}^{*},\pi_{\mbox{B}}^{*}}}({\bf
s})=\max_{\pi_{\mbox{\tiny
A}}\in\Pi}J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}^{*}}}({\bf
s})=\min_{\pi_{\mbox{\tiny
B}}\in\Pi}J^{{\tiny\pi_{\mbox{A}}^{*},\pi_{\mbox{B}}}}({\bf
s})=\underline{J}({\bf s})=\overline{J}({\bf s})=J^{*}({\bf s}).$ (3)

It follows from (3) that $\pi_{\mbox{\tiny A}}^{*}$ and $\pi_{\mbox{\tiny
B}}^{*}$ are optimal or equilibrium strategies for A and B, respectively, and
we say $J^{*}({\bf s})$ is the equilibrium value of the game starting from
state ${\bf s}$.

In our setting, the ZSG is an undiscounted stochastic shortest-path (SSP) ZSG
where the absorbing state $\tau$ (that immediately follows player A or B
winning) is rewardless and reached with probability one under all strategies
for A and B, i.e. we assume every strategy is proper. It is easy to show that
this assumption holds in our darts set-up. Under these conditions Patek and
Bertsekas (1997) showed that (3) continues to hold and that $J^{*}({\bf s})$
is the unique equilibrium of the ZSG.

Patek and Bertsekas (1997) actually considered simultaneous-move ZSGs where
the optimal equilibrium strategies are randomized. Our darts ZSG is an
alternate-move ZSG but it is easy to write this as a simultaneous-move game.
Specifically, we can imagine introducing a dummy action for each player and
assuming the dummy action is the only action available to a player when it is
in fact his opponent’s turn to throw. (Essentially, this is what we did above
when we set $\pi_{k}({\bf s}):=\emptyset$ for player $j$ when the $t$
component of ${\bf s}$ is not equal to $j$.) This dummy action has no effect
on the state of the game. Moreover, this structure where in every stage one of
the player’s feasible action sets is a singleton (dummy) action results in
equilibrium strategies that are non-randomized.

##  5 Solving the ZSG

We now describe our approach to solving the ZSG (3). As noted above, Patek and
Bertsekas (1997) showed that (3) holds in our ZSG setting and that $J^{*}({\bf
s})$ is the unique equilibrium of the game. Moreover, they also showed that
value iteration will converge to the unique optimal solution, and that is the
approach we take in this paper. Specifically, we will initialize our value
iteration by first computing player B’s optimal NS strategy $\pi_{\mbox{\tiny
B}}^{\mbox{\tiny NS}}$. We then compute A’s best-response strategy to
$\pi_{\mbox{\tiny B}}^{\mbox{\tiny NS}}$ and we call this $\pi_{\mbox{\tiny
A}}^{\mbox{\tiny BR}}$. We can then compute B’s best-response to
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny BR}}$ and iterate in this manner until
convergence.

Moreover, convergence is easy to assess since solving player A’s (B’s) best-
response to any player B (A) strategy leads to an upper (lower) bound on the
value of the game $J^{*}({\bf s})$, i.e. the win-probability for player A when
both players play their equilibrium strategies. This follows immediately from
(3). At each value iteration, we can therefore compute lower and upper bounds
on the optimal value $J^{*}({\bf s})$ and stop when these bounds are
sufficiently close to each other. In our darts setting, we find that this
typically takes only two or three iterations.

In Section 5.1 we formulate and solve the best-response problem, and then in
Section 5.2 we describe our value iteration algorithm for solving the ZSG in
more detail. Appendix B contains the NS model formulation as well as our
algorithm for computing the NS strategy.

###  5.1 The Best-Response Problem

We now solve for player A’s best-response strategy to player B’s strategy
$\pi_{\mbox{\tiny B}}\in\Pi$. We therefore assume $\pi_{\mbox{\tiny B}}$ is
fixed in this subsection. Recall that the state of the game at any time is
denoted by ${\bf s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},t,i,u)$. However,
we can avoid including $t\in\\{\mbox{A},\mbox{B}\\}$ as a component of ${\bf
s}$ by assuming it is always player A’s turn and noting that player B’s score
will change (randomly according to B’s strategy $\pi_{\mbox{\tiny B}}$) at the
end of A’s turn. With this new definition of ${\cal S}_{\mbox{\tiny ZSG}}$, we
define the state transition function for A’s best-response as

$f((s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u),z,s_{\mbox{\tiny
B}}^{\mbox{\tiny new}}):=\left\\{\begin{array}[]{ll}(0,s_{\mbox{\tiny
B}},0,0),&s_{\mbox{\tiny A}}=u+h(z)\mbox{ and }z\mbox{ a double,
}i\in\\{1,2,3\\}\\\ (s_{\mbox{\tiny A}},s_{\mbox{\tiny B}}^{\mbox{\tiny
new}},3,0),&s_{\mbox{\tiny A}}-(u+h(z))\leq 1\mbox{ and not }\\\
&\hfill(s_{\mbox{\tiny A}}=u+h(z),\ z\mbox{ a double}),i\in\\{1,2,3\\}\\\
(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i-1,u+h(z)),&s_{\mbox{\tiny
A}}>(u+h(z))+1,\ i>1\\\ (s_{\mbox{\tiny A}}-u-h(z),s_{\mbox{\tiny
B}}^{\mbox{\tiny new}},3,0),&s_{\mbox{\tiny A}}>(u+h(z))+1,\
i=1\end{array}\right.$ (4)

where $h(z)\geq 0$ is the random score of the dart. Player A wins the leg in
case 1 of (4), while in case 3 he still has at least one more throw in his
turn and has not gone bust. Player B’s updated score after A’s turn ends is
denoted by $s_{\mbox{\tiny B}}^{\mbox{\tiny new}}$, and it must satisfy $0\leq
s_{\mbox{\tiny B}}^{\mbox{\tiny new}}\leq s_{\mbox{\tiny B}}$. So
$s_{\mbox{\tiny B}}^{\mbox{\tiny new}}$ is just the result of B’s turn which
takes place between successive turns of A. We observe that B’s score updates
to $s_{\mbox{\tiny B}}^{\mbox{\tiny new}}$ only in cases 2 and 4 of (4) when A
has either gone bust (case 2) or ended his turn without winning (case 4).

Player A wins the leg and therefore obtains a reward of 1 only in the state
$(0,s_{\mbox{\tiny B}},0,0)$, i.e. in case 1 of (4), after which we assume the
state immediately moves to the rewardless absorbing state $\tau$. The dynamics
in (4) implicitly assume that $s_{\mbox{\tiny B}}^{\mbox{\tiny new}}\not=0$.
However, it should be understood that any transition where $s_{\mbox{\tiny
B}}^{\mbox{\tiny new}}=0$ implies that B wins the leg and is immediately
followed by a transition to $\tau$.

Let $J^{\mbox{\tiny A-BR}}({\bf s})$ denote player A’s best-response value
function to player B’s strategy $\pi_{\mbox{\tiny B}}\in\Pi$ for arbitrary
states ${\bf s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)\in{\cal
S}_{\mbox{\tiny ZSG}}$. Then, player A’s best-response can be formulated as
the MDP

$J^{\mbox{\tiny A-BR}}({\bf s}_{1})=\max_{\pi_{\mbox{\tiny
A}}\in\Pi}J^{{\tiny\pi_{\mbox{A}},\pi_{\mbox{B}}}}({\bf s}_{1})$ (5)

where ${\bf s}_{1}:=(501,501,3,0)$ is the initial state of the game. For ease
of exposition we are assuming A is first to throw in the leg so B starts on a
score of 501 after A’s first turn. We have also suppressed the dependence of
$J^{\mbox{\tiny A-BR}}$ on $\pi_{\mbox{\tiny B}}$ to ease the notational
burden.

Problem (5) is an SSP and standard dynamic programming (DP) techniques can be
used to show that it has a unique solution. For $s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}}>0$, it satisfies the Bellman equation

$J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)=\max_{{\bf
a}\in{\cal A}}\left\\{\text{E}\left[J^{\mbox{\tiny A-BR}}(f((s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}},i,u),z,s_{\mbox{\tiny B}}^{\mbox{\tiny
new}}))\mid(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u),{\bf
a}\right]\right\\}$ (6)

with $J^{\mbox{\tiny A-BR}}(0,s_{\mbox{\tiny B}},0,0)=1$ (A winning) and
$J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny A}},0,3,0)=0$ (B winning) and where the
expectation in (6) is taken with respect to $z$ and $s_{\mbox{\tiny
B}}^{\mbox{\tiny new}}$. Value iteration converges linearly for this problem.
In particular, for a given error tolerance $\epsilon$ under a suitably
weighted norm (see e.g. Bertsekas and Yu, 2016), we can solve (6) in $O(|{\cal
S}_{\mbox{\tiny ZSG}}|\times|{\cal A}|\times\log(1/\epsilon))$ time. (We
require $O(\log(1/\epsilon))$ value iterations and each iteration requires us
to find the optimal action in each of $|{\cal S}_{\mbox{\tiny ZSG}}|$ states.
As there are $|{\cal A}|$ possible actions to consider for each state, each
iteration therefore requires $O(|{\cal S}_{\mbox{\tiny ZSG}}|\times|{\cal
A}|)$ time.) We need $O(\max\\{|{\cal S}_{\mbox{\tiny ZSG}}|,\,|{\cal A}|\\})$
space for solving these problems.

There is a monotonic structure to the DP (6): specifically, it is clear from
the rules of the game, and indeed from the state transitions in (4), that the
$s_{\mbox{\tiny A}}$ and $s_{\mbox{\tiny B}}$ components of the state
$(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)$ are non-increasing. We can use
this monotonic structure to solve (6) more efficiently. In particular, we
proceed via Algorithm 1:

Algorithm 1 Pseudo-code for solving the best-response DP

for $s_{\mbox{\tiny B}}=2:501$ do

for $s_{\mbox{\tiny A}}=2:501$ do

Solve for $J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)$
via (6) for all $(i,u)$

end for

end for

We used policy iteration to solve the DPs inside the inner for-loop of
Algorithm 1. For each such DP, we used the previously computed optimal
policies (from earlier iterations of the for-loops) to quickly find the
optimal policy. We also experimented with using value iteration for solving
these BR problems but found it to be considerably slower than policy
iteration. (While our running-time complexity was given for value iteration,
we found policy iteration to be faster. Policy iteration is also guaranteed to
converge but we are not aware of a running-time complexity result for it that
we can use for our SSPs.)

###  5.2 Solving the Zero-Sum Game

Now that we can solve for best-response strategies, we turn our attention to
the value iteration algorithm for solving the ZSG, i.e. solving for
$J^{*}({\bf s})$ as defined in (3). As mentioned at the beginning of the
section, value iteration begins by fixing an initial strategy for one player,
e.g. $\pi_{\mbox{\tiny B}}^{\mbox{\tiny NS}}$ for player B, and finding A’s
best response $\pi_{\mbox{\tiny A}}^{\mbox{\tiny BR}}$ to this strategy. This
best-response strategy can be computed by solving (6) via Algorithm 1. We then
fix A’s strategy $\pi_{\mbox{\tiny A}}^{\mbox{\tiny BR}}$ and find B’s best-
response to $\pi_{\mbox{\tiny A}}^{\mbox{\tiny BR}}$. We continue to iterate,
alternately finding A’s best-response to B and B’s best-response to A, until
convergence. The results of Patek and Bertsekas (1997) imply that value
iteration for the ZSG converges linearly; cf. the paragraph immediately
following Prop. 4.6 in Patek and Bertsekas (1997). Linear convergence implies
$O(\log(1/\epsilon))$ such iterations will be required to solve for the
optimal ZSG value functions within an error tolerance of $\epsilon$ (again
under a suitably weighted norm).

As with the best-response DP formulation of Section 5.1, we can again take
advantage of the monotonicity structure to accelerate convergence. In
particular, we used the following algorithm:

Algorithm 2 Pseudo-code for solving the ZSG exploiting the monotonicity
structure

for $s_{\mbox{\tiny B}}=2:501$ do

for $s_{\mbox{\tiny A}}=2:501$ do

Set $\pi_{\mbox{\tiny B}}=\pi_{\mbox{\tiny B}}^{\mbox{\tiny NS}}$, ${\bf
s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},\text{A},3,0)$ # initialization

while (not converged) do

Use Alg. 1 to solve for $\pi_{\mbox{\tiny A}}$ = A’s BR to $\pi_{\mbox{\tiny
B}}$ for states in the turn starting from ${\bf s}$

Use Alg. 1 to solve for $\pi_{\mbox{\tiny B}}$ = B’s BR to $\pi_{\mbox{\tiny
A}}$ for states in the turn starting from ${\bf s}$

end while

Save $\pi_{\mbox{\tiny A}}({\bf s})$, $\pi_{\mbox{\tiny B}}({\bf s})$,
$J^{*}({\bf s})$

end for

end for

The policy iteration algorithms used to solve the BR problems inside the
while-loop of Algorithm 2 were initialized using the optimal policy from
earlier iterations of the for-loops in Algorithm 2.

##  6 Numerical Study

In this section, we present results obtained by solving our NS and ZSG models
using fitted skill models for each of the top 16 professional players in the
world during the 2019 season. We fitted these skill models using a novel data-
set that we describe below in Section 6.1. In Section 6.2 we describe the
software/hardware setup that we used for our numerical investigation, as well
as the associated run-times required for executing our algorithms. In Section
6.3 we use our solutions to quantify the value for top players of playing
strategically in a single leg and over the course of a darts match consisting
of several legs. In Section 6.4 we use a hypothetical match-play situation
involving two of the professional players to demonstrate the kind of analysis
for which our BR / ZSG solutions can be used.

###  6.1 The Data-Set

Our data-set pertains to matches that were played by the top 16 professional
players in the world during the 2019 season. For each of the 16 players we
have data of the form (TR, $z$, $n$) where $n$ is the number of darts that
were aimed at the target region (TR) and achieved a score of $z$. There are a
total of 62 possible target regions, namely the single regions S1, …, S20, the
double regions D1, …, D20, the treble regions T1, …, T20, and the single and
double bulls-eye, i.e. SB and DB. Because many regions of the dartboard are
very rarely targeted, the target regions that appear in our data-set are the
treble regions T20, T19, T18, and T17 together with all the double regions D1,
…, D20, and the double bulls-eye DB.

The possible realized value of the score $z$ depends on the target region TR
since a professional darts player will only very rarely miss his target region
TR by more than a small distance. For each treble region there are therefore 6
possible $z$ scores. In the case of TR = T20, for example, the possible values
are $z\in\\{\text{T20, S20, T5, S5, T1, S1}\\}$ because the 5 and 1 segments
of the dartboard are adjacent to the 20 segment; see Figure 1. Because we only
have data for 4 treble regions, this means a total of $6\times 4=24$ data-
vectors (TR, $z$, $n$) for each player where TR is a treble region. For each
double target region TR = D$x$ for $1\leq x\leq 20$, we have 7 possible $z$
values represented in the data-set. For example, if TR = D16 then the
corresponding possible values of $z$ are $z\in\\{\text{D16, S16, D8, S8, D7,
S7, M}\\}$ because 8 and 7 are adjacent to 16 on the dartboard and where M
denotes a “miss”, i.e. a dart that fails to score because it landed outside
the double region; again see Figure 1. When TR = DB we have 22 possible $z$’s
corresponding to {DB, SB, S1, S2, …, S20}. Because targeting the DB and some
of the double regions is relatively rare, we note that for some of the (TR,
$z$, $n$) combinations we occasionally have $n=0$.

Unfortunately, when a player targets a particular region, e.g. T20, we do not
know the exact point in the region at which he was aiming. We therefore make
the following assumption.

###### Assumption 1

Given a target region TR, the specific target that a player aims for is the
center of the target region with the center being defined as the midpoint of
the polar-coordinates defining the region.

We therefore assume the precise target location for each target region TR is
known to us. While this assumption may not always be true, absent any other
information it seems like the most natural assumption to make. Indeed it is
not difficult to imagine a player deviating from this assumption on occasion.
For example, suppose a player has two darts remaining in his turn and needs a
D5 to check out, i.e. win the game. Rather than aiming at the center of D5, he
may prefer to aim a little closer to the outer circular boundary of the D5
region on the basis that if he is going to miss D5 he would prefer to miss the
scoring region entirely rather than hit S5 which would leave him unable to
check out on his final dart of the turn. This argument does not apply to even
doubles such as D20, D18, D16, etc. since if the corresponding single is hit
then the player can still exit on the next dart in his turn. We also make two
further assumptions regarding so-called “bounce-outs” and the players’ within-
turn skill models.

###### Assumption 2

There is no possibility of a bounce-out occurring when a dart is thrown.

A “bounce-out” is a failure of the dart to actually land in the dartboard.
This can occur because of a poor throw or because the tip of the dart strikes
the wire that defines the boundaries between the different regions. With a
sufficiently sharp dart tip, however, this rarely happens and in fact only
approximately 3 in every 1,000 darts thrown by top players were bounce-outs in
the 2019 season.

###### Assumption 3

Each player has the same skill model for each throw in his turn.

It is known (see, for example, Ötting et al., 2020) that the success rate of
the first dart in a turn is generally smaller than the success rate of the
second and third throws of the turn. This can be explained by the need of the
dart-thrower to “re-calibrate” at the beginning of each turn. Unfortunately,
our data-set is not sufficiently granular to estimate a different skill model
for each throw in a turn and so we need to make Assumption 3. Of course if we
had sufficiently granular data then for each player we could fit one skill
model for the first throw in a turn and a separate skill model for the second
and third throws. These skill models could then easily be used in our models
with almost no increase in computational requirements.

Our data-set for a given player therefore takes the form $\\{({\bf
a}_{i},z_{i},n_{i})\\}_{i=1}^{p}$ where $p$ is the number of unique (TR, $z$)
combinations with $n>0$ for that player. We follow TPT and use the EM
algorithm to estimate each component of the skill model in (2) for each of the
sixteen players. The details of the EM algorithm are provided in Online
Appendix A.

In this paper, we assumed the skill model (2) with $M=6$. These six regions
are $R_{1}=\\{\mbox{T}20\\}$, $R_{2}=\\{\mbox{T}19\\}$,
$R_{3}=\\{\mbox{T}18\\}$, $R_{4}=\\{\mbox{T}17\\}$, $R_{5}=\\{\mbox{DB}\\}$
and $R_{6}=\\{\mbox{D}1,\ldots,\mbox{D}20\\}$. Only $R_{6}$ then consists of
more than one target region. This choice was largely governed by the available
data. For example, while we have a lot of data on the high trebles, i.e. T20
to T17, we have relatively less data on the doubles. To be clear, we do have a
reasonable amount of data on some particular doubles, e.g. D20 and D16, but we
have relatively little data on many of the others. Rather than fitting
individual skill models to D20 and D16 as we did with T20, T19, T18, and T17,
we instead decided to fit a single skill model to all of the doubles.

###  6.2 Software/Hardware, Run-Times and Memory Requirements

All of our numerical results were obtained using Python running on an Intel
Xeon E5-2620 v4 machine with a 2.1GHZ CPU and an Nvidia GeForce GTX 1080 Ti
GPU. The size of the state-space ${\cal S}_{\mbox{\tiny ZSG}}$ for solving a
best-response problem was on the order of $501^{2}\times(1+61+121)\approx 45$
million. ($1$, $61$ and $121$ are the number of possible $u$ values when the
number of throws $i$ remaining in a turn is $3$, $2$ and $1$, respectively.
Therefore $1+61+121$ is the number of $(i,u)$ combinations possible in a given
state.) We also needed to specify the space ${\cal A}$ of permissable actions,
i.e. targets ${\bf a}$. We adopted a brute force approach and divided the
dartboard into a grid of 1mm $\times$ 1mm squares and took ${\cal A}$ to be
the centers of each of these squares. Since the dartboard has a radius of
170mm, this resulted in an action-space with approximately $3.14\times
170^{2}=90,785$ possible targets. (We also used this 1mm $\times$ 1mm
discretization for the numerical integrations required to compute $p(z;\,{\bf
a})$ from $p(x,y;\,{\bf a})$ as discussed in Section 3.) When computing
$J^{*}({\bf s})$ for a ZSG with this action set, Algorithm 2 converged to a
relative error of $10^{-9}$ after just 2 or 3 value iterations for a majority
of the states. Each value iteration step, however, needs to examine every
possible target in ${\cal A}$ to determine a player’s best target and this
must be done for each state ${\bf s}$. As a result Algorithm 2 originally took
hundreds of hours to run when implemented naively. Through the use of
vectorization and GPU computation techniques, however, we were able to obtain
substantial improvements and ultimately solve a ZSG via Algorithm 2 in
approximately 5 hours. Online Appendix E provides further details regarding
these improvements.

We might have achieved considerable additional improvements, however, if we
took advantage of specific knowledge regarding the game of darts. For example,
early in the leg the players only ever consider aiming for T20. Similarly,
single regions are only ever considered when a player is relatively close to
checking out. These examples and many others could be used to drastically
reduce the size of the possible action spaces (by making them state-dependent)
and might help us to achieve substantial speedups.

A link to the data and Python code that we used for our numerical study is
available at https://github.com/wangchunsem/OptimalDarts.

###  6.3 Quantifying the Value of Playing Strategically

We compute the leg win-probability for each player playing their NS,
equilibrium, or BR (to NS) strategies against every other player over a single
leg. In particular, we consider the six combinations E-E N-N N-E E-N N-B B-N
where E, N, and B denote the equilibrium, NS, and BR (to NS) strategies,
respectively. So for example, E-E denotes a ZSG where both players play their
equilibrium strategies, N-E denotes a ZSG where player A plays his NS strategy
and player B plays his equilibrium strategy, and N-B denotes a ZSG where
player A plays his NS strategy and player B plays his BR to player A’s NS
strategy. The results are displayed in Tables 2 and 3 in Online Appendix B.
The main takeaway from our analysis is that the strategic aspect of the game
when played over a single leg is not substantial. Indeed, across all players
the difference in win-probability for player A when N-E is played instead of
E-E is on the order of just 0.2% to 0.6%. Put another way, if a player plays
his NS strategy (and therefore always ignores the score of his opponent)
rather than his equilibrium strategy, then the reduction in his probability of
winning the leg varies by just 0.2% to 0.6%.

These numbers, however, may misstate the value of playing strategically in the
real world where matches are played over multiple legs (or sets) rather than
just a single leg. For example, a match might easily be played as the best of
21 legs or 35 legs and a small edge in win-probability over a single leg may
translate into a considerably larger edge in win-probability over a match
consisting of multiple legs. To investigate this possibility, we will assume
for the remainder of this section that player B always plays his equilibrium
strategy $\pi_{\mbox{\tiny B}}^{*}$. We know that player A has a small
advantage on the order of approx 0.2% to 0.6% over a single leg when he plays
$\pi_{\mbox{\tiny A}}^{*}$, his equilibrium strategy, rather than
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$, his NS strategy. But what is the
cumulative advantage of A playing $\pi_{\mbox{\tiny A}}^{*}$ rather than
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ over a full match consisting of $N$
legs where the players alternate in starting each leg? To address this, let
$P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}},N)$ be the probability that A wins
such an N-leg match when A plays $\pi_{\mbox{\tiny A}}$ throughout the match
and A starts the first leg. (We could also consider the case where B starts
the first leg but the difference will be negligible for realistic values of
$N$.) We are therefore interested in evaluating A’s increase in probability of
winning the match when he plays $\pi_{\mbox{\tiny A}}^{*}$ rather than
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ against $\pi_{\mbox{\tiny B}}^{*}$.
We define this increase or gain in win-probability as

$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N):=P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{*},N)-P_{\mbox{\tiny
A}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ (7)

where we omit the dependence on $\pi_{\mbox{\tiny B}}^{*}$ as this is fixed
throughout. Each term in (7) is easily calculated as discussed in Online
Appendix C.

Table 1: $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ in a Match of $N=31$ Legs

| p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 | p11 | p12 | p13 | p14 | p15 | p16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
p1 | 0.9 | 0.9 | 0.9 | 0.3 | 1.0 | 0.9 | 0.7 | 0.9 | 0.7 | 0.9 | 0.8 | 0.9 | 1.0 | 0.8 | 0.6 | 0.9
p2 | 1.4 | 1.4 | 1.4 | 0.6 | 1.4 | 1.4 | 1.0 | 1.3 | 1.1 | 1.4 | 1.3 | 1.4 | 1.5 | 1.3 | 0.9 | 1.4
p3 | 1.1 | 1.1 | 1.1 | 0.4 | 1.2 | 1.1 | 0.9 | 1.0 | 0.7 | 1.2 | 1.0 | 1.0 | 1.1 | 1.0 | 0.6 | 1.1
p4 | 0.8 | 0.8 | 0.6 | 1.4 | 0.6 | 1.1 | 0.2 | 1.1 | 1.5 | 0.5 | 0.9 | 1.1 | 1.1 | 1.0 | 1.5 | 1.0
p5 | 1.8 | 1.8 | 1.9 | 0.6 | 2.0 | 1.7 | 1.5 | 1.6 | 1.2 | 1.9 | 1.6 | 1.7 | 1.8 | 1.6 | 1.0 | 1.8
p6 | 1.5 | 1.5 | 1.4 | 0.8 | 1.5 | 1.7 | 0.9 | 1.6 | 1.4 | 1.4 | 1.5 | 1.6 | 1.7 | 1.5 | 1.3 | 1.6
p7 | 1.2 | 1.1 | 1.3 | 0.2 | 1.3 | 0.9 | 1.5 | 0.8 | 0.5 | 1.4 | 0.9 | 0.9 | 1.0 | 0.8 | 0.4 | 1.0
p8 | 1.8 | 1.8 | 1.7 | 1.2 | 1.8 | 2.1 | 1.0 | 2.0 | 1.9 | 1.6 | 1.8 | 2.0 | 2.2 | 1.9 | 1.7 | 2.0
p9 | 0.6 | 0.6 | 0.5 | 0.6 | 0.6 | 0.8 | 0.3 | 0.7 | 0.8 | 0.5 | 0.6 | 0.7 | 0.8 | 0.7 | 0.8 | 0.7
p10 | 1.1 | 1.1 | 1.2 | 0.3 | 1.2 | 0.9 | 1.1 | 0.9 | 0.6 | 1.2 | 1.0 | 0.9 | 1.0 | 0.9 | 0.5 | 1.0
p11 | 1.1 | 1.1 | 1.1 | 0.6 | 1.1 | 1.2 | 0.7 | 1.1 | 1.0 | 1.0 | 1.1 | 1.2 | 1.3 | 1.1 | 0.9 | 1.2
p12 | 1.3 | 1.3 | 1.2 | 0.8 | 1.3 | 1.5 | 0.8 | 1.4 | 1.3 | 1.2 | 1.3 | 1.5 | 1.6 | 1.4 | 1.2 | 1.5
p13 | 0.7 | 0.7 | 0.6 | 0.4 | 0.7 | 0.7 | 0.4 | 0.7 | 0.6 | 0.6 | 0.6 | 0.7 | 0.8 | 0.7 | 0.6 | 0.7
p14 | 1.3 | 1.3 | 1.3 | 0.8 | 1.3 | 1.5 | 0.8 | 1.4 | 1.3 | 1.2 | 1.3 | 1.4 | 1.6 | 1.4 | 1.2 | 1.4
p15 | 1.1 | 1.1 | 0.9 | 1.4 | 1.0 | 1.5 | 0.4 | 1.5 | 1.8 | 0.8 | 1.2 | 1.5 | 1.6 | 1.4 | 1.8 | 1.4
p16 | 1.0 | 1.0 | 1.0 | 0.5 | 1.0 | 1.1 | 0.7 | 1.0 | 0.9 | 0.9 | 0.9 | 1.0 | 1.2 | 1.0 | 0.8 | 1.1

Note: Numbers are in percentages. The player on the left column is player A
who begins the first leg of the match. The player on the top row is player B
who uses his equilibrium strategy $\pi_{\mbox{\tiny B}}^{*}$ throughout. The
players’ names are provided in Tables 2 and 3 of Online Appendix B.

Table 1 displays $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ for all combinations of players when $N=31$ which
corresponds to a typical length match in professional tournaments. We can see
that the value of playing strategically ranges from 0.2% to 2.2% in this case.
For comparison purposes we also display in Online Appendix C the corresponding
tables for $N=1$, $21$, and $35$. For the latter two cases we see that
$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N)$ varies from 0.3% to 1.8% and from 0.2% to 2.3%, respectively. These
ranges are comparable with the $N=31$ case displayed in Table 1 and on average
are approximately two to three times higher than they are in the $N=1$ case.
While seemingly small, a value of $\mbox{Gain}(\pi_{\mbox{\tiny
A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ in the 1%-2% range is still
quite considerable given how closely matched top players are. This finding
suggests that in such matches top players would be at a considerable
disadvantage playing an NS strategy rather than an equilibrium strategy
against other top players who play their equilibrium strategies.

The function $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ typically monotonically increases in $N$ until some
value $N^{*}$ say, after which it decreases monotonically to zero. This latter
behavior is explained by the fact that $P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny
A}}^{*},N)$ and $P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$
will either both converge to 0 or both converge to 1 as $N\to\infty$ unless
playing $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ makes A an underdog but
switching to playing $\pi_{\mbox{\tiny A}}^{*}$ makes him a favorite. In this
latter case we would see $\mbox{Gain}(\pi_{\mbox{\tiny
A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ monotonically increases to
1 as $N\to\infty$. Indeed we observed that
$\lim_{N\to\infty}\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)\to 1$ in the case of Chisnall (player A) vs. Cross
(player B) who, as discussed in Online Appendix B, are very evenly matched.
This is a case where (at least in the limit of large $N$) taking account of an
opponent’s score and playing the equilibrium strategy makes all the difference
between winning and losing. For opponents as closely matched as this, however,
$N$ would need to be very large (and on the order of 1,000’s) before we would
actually see $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ get very close to 1.

###  6.4 Analysis of Matchplay Situations

Having solved for the equilibrium strategies of the ZSG, we can easily analyze
hypothetical or real-world match-play strategic situations that might occur.
These situations cannot be properly analyzed without solving the relevant ZSG
as they are inherently strategic in that the optimal action for one player
depends on the score of his opponent. Moreover, these kinds of situations are
the most interesting and dramatic situations that arise in professional
matches, since they occur near the end of a leg when both players are
relatively close in score and therefore both have a reasonable chance of
winning the leg. In Figures 2(a) and 2(b) we consider two hypothetical
situations of Anderson (player A) vs. Aspinall (player B), where they both
play their equilibrium strategies. The situation is identical in each figure
except that Aspinall is on a score of 50 in the first and a score of 150 in
the second. Anderson has one throw remaining in his turn and needs
$s_{\mbox{\tiny A}}-u=170-120=50$ to exit on this throw. In Figure 2(a)
Anderson must play aggressively and aim for the DB in which case he has a win-
probability of 46.8%. If he plays defensively and does not aim for the DB then
his win-probability will drop by between 15-20% depending on where he aims.
This is because Aspinall, on a score of 50, is very likely to exit on his next
turn. In contrast, in Figure 2(b) we see that Anderson should play defensively
and aim for S10 rather than DB. This is because Aspinall, on a score of 150,
is far less likely to exit on his next turn. We note that Anderson’s NS
strategy would aim at DB in both scenarios. This hypothetical situation serves
to emphasize the importance of playing strategically in the later part of a
leg. The dependence of the optimal strategy on the opponent’s score and his
skill model also explains why it is difficult to explain the optimal strategy
in terms of a few simple rules.

(a) Player A aims at DB in the third throw when he has 50 remaining and B is
also on 50.

(b) Player A aims at S10 in the third throw when he has 50 remaining and B is
on 150.

(c) $\pi_{\mbox{\tiny A}}^{*}$ and $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$
aim at DB.

(d) $\pi_{\mbox{\tiny A}}^{*}$ aims at S10 and $\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}}$ aims at DB.

_Note:_ Figure (a) displays a heat-map of player A’s options when it is his
turn and player B only needs 50 to exit. His optimal target is the DB and
every other target is considerably worse. In contrast, Player A is well ahead
in Figure (b) and he will play defensively by aiming at S10 instead of DB on
his third throw. Figures (c) and (d) display heat-maps $\Delta\text{P}(\mbox{A
win}):=Q^{\mbox{\tiny A-OPT}}({\bf s},{\bf
a})-J^{{\tiny\pi_{\mbox{A}}^{\mbox{\tiny NS}},\pi_{\mbox{B}}^{*}}}({\bf s})$
as a function of ${\bf a}$ for the scenarios in (a) and (b), respectively. In
(c) and (d), the blue and green crosses denote the targets chosen by
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ and $\pi_{\mbox{\tiny A}}^{*}$,
respectively.

Figure 2: Hypothetical situations of Anderson (player A) vs. Aspinall (player
B) with one throw remaining in Anderson’s turn.

Figures 2(c) and 2(d) provide an alternative view on the two scenarios
considered in Figures 2(a) and 2(b), respectively. In order to understand them
we first introduce some notation. Let $J^{{\tiny\pi_{\mbox{A}}^{\mbox{\tiny
NS}},\pi_{\mbox{B}}^{*}}}({\bf s})$ denote the probability of player A winning
from state ${\bf s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},t,i,u)$ when player
A uses $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ and player B uses his
equilibrium strategy $\pi_{\mbox{\tiny B}}^{*}$. Let $Q^{\mbox{\tiny
A-OPT}}({\bf s},{\bf a})$ denote the probability of player A winning from
state ${\bf s}$ if he aims at ${\bf a}$ on his current throw, plays his
equilibrium strategy $\pi_{\mbox{\tiny A}}^{*}$ thereafter, and player B
always plays $\pi_{\mbox{\tiny B}}^{*}$. We can therefore interpret
$Q^{\mbox{\tiny A-OPT}}({\bf s},{\bf a})$ as a so-called Q-value for player A.
We let $\Delta\text{P}(\mbox{A win}):=Q^{\mbox{\tiny A-OPT}}({\bf s},{\bf
a})-J^{{\tiny\pi_{\mbox{A}}^{\mbox{\tiny NS}},\pi_{\mbox{B}}^{*}}}({\bf s})$
which is a function of ${\bf a}$. When ${\bf a}$ is the target chosen by
$\pi_{\mbox{\tiny A}}^{*}$ (denoted by the green crosses in Figures 2(c) and
2(d)), then $\Delta\text{P}(\mbox{A win})$ is simply the difference in win-
probabilities for player A playing $\pi_{\mbox{\tiny A}}^{*}$ instead of
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ against player B playing
$\pi_{\mbox{\tiny B}}^{*}$.

In Figure 2(c) the optimal ${\bf a}$ is in the DB region and coincides with
the action chosen by $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$. Hence the
largest value of $\Delta\text{P}(\mbox{A win})$ is very small and close to
zero. (Note that $\max_{{\bf a}}\Delta\text{P}(\mbox{A win})\geq 0$ with
equality occurring only if $\pi_{\mbox{\tiny A}}^{*}=\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}}$ in state ${\bf s}$ and all possible successor states to
${\bf s}$.) In contrast, we see from Figure 2(d) that the loss in win-
probability if player A targets the DB (as recommended by $\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}}$) rather than S10 (as recommended by $\pi_{\mbox{\tiny
A}}^{*}$) is 1.8% which is quite considerable.

We emphasize that this particular situation is not pathological in any sense.
Indeed it is easy to find many situations, i.e. states ${\bf s}$, where the
difference in win-probabilities between following $\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}}$ rather than $\pi_{\mbox{\tiny A}}^{*}$ is on the order
of 1-2%. These situations ultimately come down to $\pi_{\mbox{\tiny A}}^{*}$
accepting a slight increase in the expected number of turns to exit (which is
what $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ minimizes) for a slight increase
in the probability of exiting in just one turn.

Finally, we remark that it would be disastrous for Anderson to aim at any of
T17, T18, T19, or T20 since any of these scores would result in his going bust
and therefore reverting to a score of 170 for his next turn. This would result
in a massive drop in his win-probability and this is also clear from the heat-
maps in Figure 2.

##  7 Conclusions and Further Research

In this paper we formulated and solved the game of darts as a dynamic ZSG. We
also estimated individual skill models for the top 16 professional players in
the world during the 2019 season. Using the fitted skill models and our ZSG
problem formulation and solution, we quantified the importance of playing
strategically when computing an optimal strategy. For top professionals, we
found that playing strategically (as opposed to non-strategically) results in
an increase in win-probability of just 0.2% - 0.6% over a single leg but as
much as 2.3% over a best-of-35 legs match.

There are several directions for future research. First, if time-series data
on the player’s dart-throws was available then we could try to model the form
of players as it varies through time. Sufficiently granular data would also
allow us to estimate a separate skill model for the first throw in a turn
which is typically less accurate than the second and third throws. We discuss
this further in Online Appendix D.2. Following Ötting et al. (2020), it might
also be of interest to model the within-turn hot-hand phenomenon and again,
given sufficiently granular data, this would be easy to handle in our ZSG
setting and we provide a formulation for this in Online Appendix D.2. Modeling
all of these features would therefore allow us to estimate a more accurate and
state-dependent skill model for each player.

Given a more accurate skill-model, we could then use time-series data that
included the dart-by-dart scoring evolution of matches, to estimate just how
far the top players are deviating from their (estimated) optimal strategies.
For each player we could then estimate the loss in win-probability that
results from playing sub-optimally. We note there are two possible reasons for
playing sub-optimally: (i) a player not having an accurate assessment of his
skill model and (ii) playing a sub-optimal strategy for his assumed skill
strategy and we suspect both reasons lead to sub-optimal play on occasion.
Even with the limitations of our data-set, it would be straightforward to
quantify the loss in win-probability due to (i). In particular, we could for
example assume skill model (2) was the “true” model while the player
mistakenly assumes model (1) is correct. It would then be straightforward to
assess how much is lost in win-probability while playing against other top
players who do know their correct skill models and play their corresponding
equilibrium strategies.

Another interesting direction for future work is in modifying the rules of
darts so that the value of strategic playing, i.e. basing your strategy on
your opponent’s current score, becomes more important. TPT did some related
analysis on this when they used a Metropolis-Hastings algorithm to search for
configurations of the dartboard segments (1 to 20) that were more challenging
for a darts player trying to maximize his expected score. Similarly, a
relatively recent article in The Guardian newspaper (Bellos, 2013) discusses
some other developments in this direction. Given the world-wide growth in the
popularity of darts, we suspect variations on the game of 501 will be
investigated and perhaps played in the future. In order to fully assess their
impact on the professional game, skill models and the ability to solve the
ZSG’s will be required.

## Acknowledgements

We are very grateful to Christopher Kempf (Twitter @ochepedia), Statistical
Analyst for the PDC for providing us with the data-set we used in Section 6
and for some very insightful conversations. We are also grateful to the Area
Editor, Nicola Secomandi, and the anonymous Associate Editor and referees for
their helpful comments and suggestions. All errors are our own.

##  Appendix A Dartboard Geometry

A standard competition dartboard has the following measurements (in
millimeters):

Distance | Measurement
---|---
Center to DB wire | 6.35
Center to SB wire | 15.9
Center to inner triple wire | 99
Center to outer triple wire | 107
Center to inner double wire | 162
Center to outer double wire | 170

Given these measurements and taking the center of DB to be $(0,0)$, it is
straightforward to compute the map $z=g(x,y)$ that maps any $(x,y)$-coordinate
on the dartboard to the corresponding dart score.

##  Appendix B The Non-Strategic Model

In this appendix we formulate the NS problem where a player’s goal is simply
to get to zero (by checking out on a double) as quickly as possible. We will
formulate this as an MDP problem.

The state space ${\cal S}_{\mbox{\tiny NS}}$ for the MDP consists of all
triples ${\bf s}=(s,i,u)$ where $s$ is the score at the beginning of the turn,
$i\in\\{1,2,3\\}$ is the number of throws remaining in the turn and $u$ is the
score-to-date within the turn. We therefore have $i=3$ and $u=0$ at the start
of each turn. The state transition function $f((s,i,u),z)$ depends on the
current state ${\bf s}$ and the realized dart score $z$. It is defined as
follows.

$f((s,i,u),z):=\left\\{\begin{array}[]{ll}(0,0,0),&s=u+h(z)\mbox{ and }z\mbox{
a double, }i\in\\{1,2,3\\}\\\ (s,3,0),&s-(u+h(z))\leq 1\\\ &\hfill\mbox{ and
not }(s=u+h(z),\ z\mbox{ a double}),i\in\\{1,2,3\\}\\\
(s,i-1,u+h(z)),&s>(u+h(z))+1,\ i>1\\\ (s-u-h(z),3,0),&s>(u+h(z))+1,\
i=1.\end{array}\right.$ (B-1)

The first two cases in (B-1) correspond to checking out and going bust,
respectively. The third case corresponds to neither going bust nor checking
out and still having at least one throw remaining in the turn. The final case
corresponds to the last throw of a turn and neither checking out nor going
bust.

We can model the per-stage costs of this MDP according to
$c(s,i,u):=1_{\\{i=3\\}}$, so that we incur a cost of 1 at the beginning of
each turn. Recall that the action-space ${\cal A}$ is the finite set of
possible targets on the dartboard. It is state-independent.

An admissible policy $\pi:=\\{\pi_{1},\pi_{2},\ldots\\}$ is a policy where
each $\pi_{k}$ is a decision rule defining the target of the $k^{th}$ dart.
That is, $\pi_{k}({\bf s})\in{\cal A}$ for any ${\bf s}\in{\cal
S}_{\mbox{\tiny NS}}$ and with $k\in\mathbb{N}^{+}$ denoting the $k^{th}$
throw of the dart (across all turns). The total expected cost for any policy
$\pi$ is then defined as $V_{\pi}({\bf
s}):=\lim_{K\to\infty}\text{E}\left[\sum_{k=1}^{K}c({\bf s}_{k})\mid{\bf
s}_{1}={\bf s}\right]$ where $K$ denotes the total number of permitted dart
throws in a finite-horizon version of the problem. Letting $\Pi$ represent the
space of all admissible policies, the MDP problem is to solve for

$V_{\mbox{\tiny NS}}({\bf s}_{1}):=\min_{\pi\in\Pi}V_{\pi}({\bf s}_{1})$ (B-2)

where ${\bf s}_{1}:=(501,3,0)$ is the initial state of the game. We recognize
this infinite-horizon MDP as a stochastic shortest-path, i.e. SSP, problem
where the goal is to check out in as few turns as possible. A stationary
policy $\pi$ is a policy where the decision rule $\pi_{k}$ does not depend on
$k$. In this case we will use $\pi$ to denote both the policy and the decision
rule $\pi({\bf s})$. It is well known (see e.g. Bertsekas, 2012) that there
exists an optimal stationary policy under fairly modest assumptions. One such
assumption, which is sufficient but not necessary, is that every policy is
proper, i.e. every policy will terminate with probability one. It is easy to
argue that this assumption holds in our darts set-up.

A non-strategic model of a darts leg has previously been proposed and solved
by Baird (2020). Kohler (1982) solved the same model except he ignored the
turn feature. In particular, he assumed a player’s score after going bust was
his score immediately prior to the dart on which he went bust rather than his
score at the beginning of the turn. (For the sake of completion we formulate
the MDP solved by Kohler (1982) in Online Appendix D.1.)

###  B.1 Solving the Non-Strategic Problem

Chapter 2 of Bertsekas (2012) implies that the actions returned by the
decision rules of an optimal stationary policy for (B-2) satisfy the Bellman
equation

$\displaystyle V_{\mbox{\tiny NS}}(s,i,u)$ $\displaystyle=$
$\displaystyle\min_{{\bf a}\in{\cal
A}}\left\\{\text{E}[1_{\\{i=3\\}}+V_{\mbox{\tiny
NS}}(f((s,i,u),z))\mid(s,i,u),{\bf a}]\right\\}$ (B-3) $\displaystyle=$
$\displaystyle\min_{{\bf a}\in{\cal
A}}\left\\{\sum_{z}\left[1_{\\{i=3\\}}+V_{\mbox{\tiny
NS}}(f((s,i,u),z))\right]\,p(z;\,{\bf a})\right\\}$

for every feasible state $(s,i,u)$ with $V_{\mbox{\tiny NS}}(0,0,0):=0$.

As with our best-response problem (6), there is a monotonic structure to this
DP (B-3): specifically, it is clear from the rules of the game, and indeed
from the state transitions in (B-1), that the $s$ component of the state
$(s,i,u)$ is non-increasing. We can therefore use this to successively solve
for $V_{\mbox{\tiny NS}}(2,i,u)$, $V_{\mbox{\tiny NS}}(3,i,u)$, $\ldots$,
$V_{\mbox{\tiny NS}}(501,i,u)$ for all $i,u$. If we let $\pi^{\mbox{\tiny
NS}}$ denote the corresponding optimal strategy then $\pi^{\mbox{\tiny
NS}}(s,i,u)$ is the optimal decision rule defining where the non-strategic
(NS) dart thrower should aim when his current state is $(s,i,u)$. While the
solution of SSP problems is still an active research area (e.g. see Bertsekas
and Yu, 2016), our finite state and action spaces together with the assumption
of policies being proper are sufficient to guarantee linear convergence of
value iteration for solving (B-3). In particular, for a given error tolerance
$\epsilon$ under a suitably weighted norm (see e.g. Bertsekas and Yu, 2016),
we can solve (B-2) in $O(|{\cal S}_{\mbox{\tiny NS}}|\times|{\cal
A}|\times\log(1/\epsilon))$ time. (We require $O(\log(1/\epsilon))$ value
iterations and each iteration requires us to find the optimal action in each
of $|{\cal S}_{\mbox{\tiny NS}}|$ states. As there are $|{\cal A}|$ possible
actions to consider for each state, each iteration therefore requires
$O(|{\cal S}_{\mbox{\tiny NS}}|\times|{\cal A}|)$ time.) We need
$O(\max\\{|{\cal S}_{\mbox{\tiny NS}}|,\,|{\cal A}|\\})$ space for solving
these problems.

In our numerical experiments, we took advantage of the aforementioned
monotonicity structure and used policy iteration rather than value iteration
for solving the SSPs. (While our running-time complexity was given for value
iteration, as with the best-response problem for the ZSG, we again found
policy iteration to be faster than value iteration.)

###  B.2 Evaluating $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ and
$\pi_{\mbox{\tiny B}}^{\mbox{\tiny NS}}$ Going Head-to-Head

Suppose now that players A and B both play their NS strategies
$\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$ and $\pi_{\mbox{\tiny
B}}^{\mbox{\tiny NS}}$, respectively, as determined by (B-3). These strategies
will be different in general as they will have different skill models and
hence different $p(z;\,{\bf a})$ distributions. It is straightforward to
compute numerically the probability of player A winning a darts leg when both
players play their respective NS strategies. This can be done recursively
which provides an effective way of solving the linear system of equations to
compute the probability of A winning before B. The recursion is very similar
to the recursion required for computing a player’s best response (see (6) in
Section 5.1) except there is no maximization required as the strategies are
fixed.

Even though the players are playing their NS strategies (which were computed
using the state-space ${\cal S}_{\mbox{\tiny NS}}$), when we evaluate them
going head-to-head we must use the state-space ${\cal S}_{\mbox{\tiny ZSG}}$
since the win-probabilities will be a function of the expanded state ${\bf
s}=(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},t,i,u)\in{\cal S}_{\mbox{\tiny
ZSG}}$. In order to compute these value functions, i.e. win-probabilities, we
will need to compute $p^{\mbox{\tiny turn}}(s,s^{\prime};\pi)$, the
probability of a player’s score moving from $s$ to $s^{\prime}\leq s$ over the
course of a single turn when he plays strategy $\pi$.

It should be clear that exactly the same approach can be used to evaluate
arbitrary strategies $\pi_{\mbox{\tiny A}}$ and $\pi_{\mbox{\tiny B}}$ (e.g.
best-response, equilibrium strategies) going head-to-head against each other.

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Online Appendix: Play Like the Pros? Solving the Game of Darts as a Dynamic
Zero-Sum Game

##  Online Appendix A Fitting the Skill Models via the EM Algorithm

We wish to fit each component of the normal skill model (2) for each of the 16
players. (Recall for each player we use six components in our model - one each
for T20, T19, T18, T17, and DB, and one for the twenty doubles D1,…,D20
combined.) We will fix a particular component and let the data for this
component be $\\{({\bf a}_{i},z_{i},n_{i})\\}_{i=1}^{p}$. The log-likelihood
then satisfies

$l({{\bm{\Sigma}}};\,\\{({\bf
a}_{i},z_{i},n_{i})\\}_{i=1}^{p})\propto\sum_{i=1}^{p}n_{i}\,\log\left(\text{P}\left((x,y)\in
R(z_{i})\mid(x,y)\sim\mbox{N}_{2}\left({\bf
a}_{i},{{\bm{\Sigma}}}\right)\right)\right)$ (A-4)

where $R(z_{i})$ is the region defined by the dart $z_{i}$, i.e.
$R(z_{i})=\\{(x,y)\in\mathbb{R}^{2}\,:\,g(x,y)=z_{i}\\}$, and $p$ is the
number of regions in the data-set where darts landed. So for example, if we
are fitting the skill model for any of the four treble components then $p=6$.
Further details are provided in Section 6.1. For the DB component $p=22$ and
for the remaining doubles component we have $p=20\times 7=140$. Omitting its
dependence on the data, we note that $l({{\bm{\Sigma}}})$ can be easily
computed via numerical integration but rather than attempting to maximize it
w.r.t. ${{\bm{\Sigma}}}$ directly, we can instead use the EM algorithm as
suggested by TPT.

Let $n=n_{1}+\cdots+n_{p}$ denote the total number of darts and let
${\bm{\nu}}_{i}\in\mathbb{R}^{2}$ denote the unobserved realized position of
the $i^{th}$ dart. The complete-data log-likelihood then satisfies

$\displaystyle l_{\mbox{\tiny com}}({{\bm{\Sigma}}};\,\\{({\bf
a}_{i},{\bm{\nu}}_{i},z_{i},)\\}_{i=1}^{n})$ $\displaystyle=$
$\displaystyle-\frac{n}{2}\log|{{\bm{\Sigma}}}|-\frac{1}{2}\sum_{i=1}^{n}\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)^{\top}{{\bm{\Sigma}}}^{-1}\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)$ (A-5) $\displaystyle=$
$\displaystyle-\frac{n}{2}\log|{{\bm{\Sigma}}}|-\frac{1}{2}\mbox{tr}\left({{\bm{\Sigma}}}^{-1}\sum_{i=1}^{n}\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)^{\top}\right)$

where $z_{i}=g({\bm{\nu}}_{i})$ for $i=1,\ldots,n$ and $\mbox{tr}(\cdot)$
denotes the trace operator. (There is a slight abuse of notation here as
$({\bf a}_{i},z_{i})$ now represents the target/outcome of the $i^{th}$ dart
rather than the $i^{th}$ $(TR,z)$ combination in the data-set of which there
are only $p$. The particular interpretation we have in mind should be clear
from the context. For example, in going from (A-7) to (A-8) we move from the
former interpretation to the latter one.) The E-step of the EM algorithm
requires the calculation of

$\text{E}_{{{\bm{\Sigma}}}_{\mbox{\tiny old}}}\left[l_{\mbox{\tiny
com}}({{\bm{\Sigma}}})\mid
z\right]=-\frac{n}{2}\log|{{\bm{\Sigma}}}|-\frac{1}{2}\mbox{tr}\left({{\bm{\Sigma}}}^{-1}\sum_{i=1}^{n}\text{E}_{{{\bm{\Sigma}}}_{0}}\left[\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)\left({\bf a}_{i}-{\bm{\nu}}_{i}\right)^{\top}\mid
z_{i}\right]\right)$ (A-6)

where $\text{E}_{{{\bm{\Sigma}}}_{\mbox{\tiny old}}}[\cdot\mid z]$ denotes the
expectation operator conditional on the observed dart scores
$z_{1},\ldots,z_{n}$, and using the current parameter estimate
${{\bm{\Sigma}}}_{\mbox{\tiny old}}$. The M-step requires the maximization of
(A-6) over ${{\bm{\Sigma}}}$. This is identical to the usual maximum
likelihood estimation for a multivariate Gaussian over an unknown covariance
matrix and therefore has solution

$\displaystyle{{\bm{\Sigma}}}_{\mbox{\tiny new}}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\text{E}_{{{\bm{\Sigma}}}_{\mbox{\tiny
old}}}\left[\left({\bf a}_{i}-{\bm{\nu}}_{i}\right)\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)^{\top}\mid z_{i}\right]$ (A-7) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{p}n_{i}\text{E}_{{{\bm{\Sigma}}}_{\mbox{\tiny
old}}}\left[\left({\bf a}_{i}-{\bm{\nu}}_{i}\right)\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)^{\top}\mid z_{i}\right].$ (A-8)

The expectation in (A-8) cannot be computed in closed form and so we follow
the approach of TPT and compute them numerically via importance-sampling (IS).
Specifically, we compute the expectation in (A-8) as

$\text{E}_{{{\bm{\Sigma}}}_{\mbox{\tiny old}}}\left[\left({\bf
a}_{i}-{\bm{\nu}}_{i}\right)\left({\bf a}_{i}-{\bm{\nu}}_{i}\right)^{\top}\mid
z_{i}\right]\approx\sum_{j=1}^{m}w_{i,j}\left({\bf
a}_{i}-{\bm{\nu}}_{i,j}\right)\left({\bf
a}_{i}-{\bm{\nu}}_{i,j}\right)^{\top}$ (A-9)

where $m$ is the number of samples we used, the ${\bm{\nu}}_{i,j}$’s are
samples of ${\bm{\nu}}_{i}$ and the $w_{i,j}$’s are corresponding IS weights.
(We used $m=5,000$ when fitting our skill models.) Specifically

$w_{i,j}=\frac{p_{i,j}/q_{i,j}}{\sum_{j=1}^{m}p_{i,j}/q_{i,j}}$ (A-10)

where $p_{i,j}$ is the density of ${\bm{\nu}}_{i,j}$ under the true
distribution of ${\bm{\nu}}_{i}$ conditional on $z_{i}$ and using the
parameter estimate ${{\bm{\Sigma}}}_{\mbox{\tiny old}}$, and $q_{i,j}$ is the
sampling density of ${\bm{\nu}}_{i,j}$. Because we only know $p_{i,j}$ up to a
multiplicative constant, we normalize in (A-10) to ensure the weights sum to
1. TPT suggested taking the sampling distribution $q_{i,j}$ to be uniform on
$R(z_{i})$ and this of course satisfies the IS requirement that $p=0$ whenever
$q=0$. Moreover with this choice (A-10) is easy to compute since by assumption
$q_{i,j}\propto 1$ and $p_{i,j}$ is the bivariate normal PDF with mean ${\bf
a}_{i}$ and variance-covariance matrix ${{\bm{\Sigma}}}_{\mbox{\tiny old}}$
evaluated at ${\bm{\nu}}_{i,j}$. The EM algorithm then proceeds by iterating
(A-8) to (A-10) until convergence.

##  Online Appendix B Leg Win-Probability Results

In this online appendix we display the win-probability results when each
possible pair of opponents play the various strategies against each other.
Tables 2 and 3 display the leg win-probability for each player playing their
non-strategic (NS), equilibrium, or BR (to NS) strategies against every other
player.

Table 2: Probability of Winning a Single Darts Leg

| p1 (Anderson) | p2 (Aspinall) | p3 (Chisnall) | p4 (Clayton)
---|---|---|---|---
E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N
p1 | 64.0 | 64.0 | 63.8 | 64.2 | 63.8 | 64.2 | 64.2 | 64.2 | 64.0 | 64.5 | 64.0 | 64.5 | 62.6 | 62.6 | 62.4 | 62.8 | 62.4 | 62.8 | 73.4 | 73.6 | 73.2 | 73.7 | 73.2 | 73.7
p2 | 63.9 | 63.6 | 63.5 | 64.0 | 63.5 | 64.0 | 64.1 | 63.9 | 63.7 | 64.3 | 63.7 | 64.3 | 62.4 | 62.2 | 62.1 | 62.6 | 62.1 | 62.6 | 73.3 | 73.3 | 73.0 | 73.7 | 73.0 | 73.7
p3 | 65.8 | 65.6 | 65.4 | 65.9 | 65.4 | 65.9 | 65.9 | 65.8 | 65.6 | 66.2 | 65.6 | 66.2 | 64.4 | 64.2 | 64.1 | 64.5 | 64.1 | 64.5 | 74.9 | 74.9 | 74.6 | 75.2 | 74.6 | 75.2
p4 | 51.5 | 51.3 | 51.2 | 51.7 | 51.2 | 51.7 | 51.8 | 51.6 | 51.4 | 52.0 | 51.4 | 52.0 | 50.0 | 49.8 | 49.7 | 50.2 | 49.7 | 50.2 | 62.0 | 62.0 | 61.7 | 62.3 | 61.7 | 62.3
p5 | 65.9 | 65.5 | 65.4 | 66.1 | 65.4 | 66.1 | 66.1 | 65.8 | 65.6 | 66.3 | 65.6 | 66.3 | 64.5 | 64.1 | 64.0 | 64.6 | 63.9 | 64.6 | 75.2 | 75.1 | 74.8 | 75.6 | 74.8 | 75.6
p6 | 61.0 | 60.8 | 60.7 | 61.2 | 60.7 | 61.2 | 61.2 | 61.1 | 60.9 | 61.5 | 60.9 | 61.5 | 59.5 | 59.3 | 59.2 | 59.7 | 59.2 | 59.7 | 71.1 | 71.1 | 70.8 | 71.4 | 70.8 | 71.4
p7 | 70.9 | 70.6 | 70.4 | 71.0 | 70.4 | 71.0 | 71.0 | 70.8 | 70.6 | 71.3 | 70.6 | 71.3 | 69.6 | 69.3 | 69.1 | 69.7 | 69.1 | 69.7 | 79.2 | 79.2 | 78.8 | 79.5 | 78.8 | 79.5
p8 | 60.7 | 60.3 | 60.1 | 60.8 | 60.1 | 60.8 | 60.9 | 60.6 | 60.4 | 61.1 | 60.4 | 61.1 | 59.2 | 58.8 | 58.6 | 59.4 | 58.6 | 59.4 | 70.6 | 70.4 | 70.1 | 70.9 | 70.1 | 70.9
p9 | 56.4 | 56.4 | 56.2 | 56.6 | 56.2 | 56.6 | 56.7 | 56.7 | 56.5 | 56.9 | 56.5 | 56.9 | 54.9 | 54.9 | 54.7 | 55.1 | 54.7 | 55.1 | 66.8 | 67.0 | 66.7 | 67.2 | 66.7 | 67.2
p10 | 67.2 | 67.1 | 66.9 | 67.4 | 66.9 | 67.4 | 67.4 | 67.3 | 67.1 | 67.6 | 67.1 | 67.6 | 65.8 | 65.7 | 65.5 | 66.0 | 65.5 | 66.0 | 76.2 | 76.3 | 75.9 | 76.5 | 75.9 | 76.5
p11 | 62.3 | 62.1 | 61.9 | 62.4 | 61.9 | 62.4 | 62.5 | 62.4 | 62.2 | 62.7 | 62.2 | 62.7 | 60.8 | 60.7 | 60.5 | 61.0 | 60.5 | 61.0 | 71.8 | 71.8 | 71.5 | 72.1 | 71.5 | 72.1
p12 | 60.9 | 60.7 | 60.5 | 61.1 | 60.5 | 61.1 | 61.1 | 61.0 | 60.8 | 61.4 | 60.8 | 61.4 | 59.5 | 59.2 | 59.1 | 59.6 | 59.1 | 59.6 | 70.8 | 70.8 | 70.5 | 71.1 | 70.5 | 71.1
p13 | 61.6 | 61.6 | 61.4 | 61.8 | 61.4 | 61.8 | 61.8 | 61.9 | 61.7 | 62.1 | 61.7 | 62.1 | 60.1 | 60.1 | 59.9 | 60.3 | 59.9 | 60.3 | 71.7 | 71.8 | 71.5 | 72.0 | 71.5 | 72.0
p14 | 61.0 | 60.7 | 60.6 | 61.1 | 60.6 | 61.1 | 61.2 | 61.0 | 60.8 | 61.4 | 60.8 | 61.4 | 59.5 | 59.3 | 59.1 | 59.7 | 59.1 | 59.7 | 70.7 | 70.7 | 70.4 | 71.0 | 70.4 | 71.0
p15 | 54.9 | 54.6 | 54.5 | 55.1 | 54.5 | 55.1 | 55.2 | 55.0 | 54.7 | 55.4 | 54.7 | 55.4 | 53.4 | 53.1 | 52.9 | 53.6 | 52.9 | 53.6 | 65.6 | 65.5 | 65.2 | 65.9 | 65.2 | 65.9
p16 | 62.3 | 62.2 | 62.0 | 62.5 | 62.0 | 62.5 | 62.5 | 62.4 | 62.2 | 62.7 | 62.2 | 62.7 | 60.8 | 60.7 | 60.5 | 61.0 | 60.5 | 61.0 | 72.0 | 72.1 | 71.8 | 72.3 | 71.8 | 72.3
| p5 (Cross) | p6 (Cullen) | p7 (van Gerwen) | p8 (Gurney)
E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N
p1 | 63.1 | 63.2 | 62.9 | 63.4 | 62.9 | 63.4 | 67.1 | 67.2 | 66.9 | 67.4 | 66.9 | 67.4 | 58.0 | 57.9 | 57.7 | 58.2 | 57.7 | 58.2 | 67.2 | 67.3 | 67.0 | 67.5 | 67.0 | 67.5
p2 | 62.9 | 62.8 | 62.5 | 63.2 | 62.5 | 63.2 | 66.9 | 66.9 | 66.6 | 67.3 | 66.6 | 67.3 | 57.7 | 57.6 | 57.4 | 57.9 | 57.4 | 57.9 | 67.0 | 66.9 | 66.7 | 67.3 | 66.6 | 67.3
p3 | 64.9 | 64.8 | 64.5 | 65.2 | 64.5 | 65.2 | 68.8 | 68.8 | 68.5 | 69.1 | 68.5 | 69.1 | 59.7 | 59.6 | 59.4 | 60.0 | 59.4 | 60.0 | 68.9 | 68.8 | 68.5 | 69.1 | 68.5 | 69.1
p4 | 50.3 | 50.2 | 49.9 | 50.6 | 49.9 | 50.6 | 54.6 | 54.4 | 54.2 | 54.8 | 54.2 | 54.8 | 45.3 | 45.1 | 44.9 | 45.5 | 44.9 | 45.5 | 54.8 | 54.7 | 54.4 | 55.1 | 54.4 | 55.1
p5 | 65.0 | 64.7 | 64.4 | 65.3 | 64.4 | 65.3 | 69.0 | 68.8 | 68.5 | 69.3 | 68.5 | 69.3 | 59.7 | 59.4 | 59.2 | 59.9 | 59.2 | 59.9 | 69.1 | 68.9 | 68.6 | 69.4 | 68.6 | 69.4
p6 | 60.0 | 59.9 | 59.6 | 60.3 | 59.6 | 60.3 | 64.2 | 64.2 | 63.8 | 64.5 | 63.8 | 64.6 | 54.6 | 54.5 | 54.3 | 54.9 | 54.3 | 54.9 | 64.3 | 64.3 | 64.0 | 64.6 | 63.9 | 64.6
p7 | 70.1 | 69.9 | 69.7 | 70.4 | 69.7 | 70.4 | 73.8 | 73.6 | 73.3 | 74.1 | 73.3 | 74.1 | 65.1 | 64.9 | 64.7 | 65.3 | 64.7 | 65.3 | 73.8 | 73.6 | 73.4 | 74.1 | 73.4 | 74.1
p8 | 59.6 | 59.4 | 59.1 | 59.9 | 59.1 | 59.9 | 63.8 | 63.5 | 63.3 | 64.1 | 63.2 | 64.1 | 54.4 | 54.1 | 53.8 | 54.6 | 53.8 | 54.6 | 63.9 | 63.7 | 63.4 | 64.2 | 63.4 | 64.2
p9 | 55.3 | 55.4 | 55.1 | 55.6 | 55.1 | 55.6 | 59.6 | 59.7 | 59.4 | 59.9 | 59.4 | 59.9 | 50.0 | 50.1 | 49.8 | 50.3 | 49.8 | 50.3 | 59.7 | 59.9 | 59.5 | 60.1 | 59.5 | 60.1
p10 | 66.3 | 66.3 | 66.0 | 66.6 | 66.0 | 66.6 | 70.2 | 70.2 | 69.9 | 70.6 | 69.9 | 70.6 | 61.2 | 61.1 | 60.9 | 61.4 | 60.9 | 61.4 | 70.3 | 70.3 | 70.0 | 70.6 | 70.0 | 70.6
p11 | 61.3 | 61.2 | 60.9 | 61.6 | 60.9 | 61.6 | 65.3 | 65.3 | 65.0 | 65.6 | 65.0 | 65.6 | 56.2 | 56.0 | 55.8 | 56.4 | 55.8 | 56.4 | 65.4 | 65.4 | 65.1 | 65.7 | 65.1 | 65.7
p12 | 59.9 | 59.8 | 59.5 | 60.2 | 59.5 | 60.2 | 64.1 | 64.0 | 63.7 | 64.4 | 63.7 | 64.4 | 54.6 | 54.5 | 54.2 | 54.9 | 54.2 | 54.9 | 64.2 | 64.1 | 63.8 | 64.5 | 63.8 | 64.5
p13 | 60.5 | 60.7 | 60.3 | 60.9 | 60.3 | 60.9 | 64.8 | 64.9 | 64.6 | 65.2 | 64.6 | 65.2 | 55.2 | 55.3 | 55.0 | 55.5 | 55.0 | 55.5 | 64.9 | 65.1 | 64.7 | 65.2 | 64.7 | 65.2
p14 | 59.9 | 59.8 | 59.5 | 60.2 | 59.5 | 60.2 | 64.1 | 64.0 | 63.7 | 64.4 | 63.7 | 64.4 | 54.8 | 54.6 | 54.4 | 55.0 | 54.4 | 55.0 | 64.2 | 64.1 | 63.8 | 64.5 | 63.8 | 64.5
p15 | 53.7 | 53.6 | 53.2 | 54.0 | 53.2 | 54.1 | 58.1 | 57.9 | 57.6 | 58.4 | 57.6 | 58.4 | 48.5 | 48.3 | 48.0 | 48.7 | 48.0 | 48.7 | 58.3 | 58.2 | 57.8 | 58.6 | 57.8 | 58.6
p16 | 61.3 | 61.3 | 61.0 | 61.6 | 61.0 | 61.6 | 65.4 | 65.4 | 65.1 | 65.7 | 65.1 | 65.7 | 56.0 | 56.0 | 55.8 | 56.3 | 55.8 | 56.3 | 65.5 | 65.5 | 65.2 | 65.8 | 65.2 | 65.8

Note: Numbers are in percentages and the player from the left column throws
first while the player (name in parentheses) from the top row throws second. N
denotes the NS strategy, E denotes the equilibrium strategy, and B denotes the
best-response strategy to an opponent’s NS strategy. We note that, to 1
decimal place, N-B and B-N are indistinguishable from N-E and E-N,
respectively, in the vast majority of cases.

Table 3: Probability of Winning a Single Darts Leg Continued

| p9 (Lewis) | p10 (Price) | p11 (Smith) | p12 (Suljovic)
---|---|---|---|---
E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N
p1 | 70.3 | 70.3 | 70.1 | 70.5 | 70.1 | 70.5 | 61.4 | 61.4 | 61.2 | 61.6 | 61.2 | 61.6 | 65.4 | 65.3 | 65.1 | 65.5 | 65.1 | 65.5 | 66.9 | 66.9 | 66.6 | 67.1 | 66.6 | 67.1
p2 | 70.2 | 70.0 | 69.8 | 70.4 | 69.8 | 70.4 | 61.2 | 61.0 | 60.8 | 61.4 | 60.8 | 61.4 | 65.2 | 65.0 | 64.8 | 65.3 | 64.8 | 65.3 | 66.7 | 66.5 | 66.3 | 66.9 | 66.3 | 66.9
p3 | 71.9 | 71.8 | 71.6 | 72.1 | 71.6 | 72.1 | 63.2 | 63.0 | 62.8 | 63.4 | 62.8 | 63.4 | 67.0 | 66.9 | 66.7 | 67.2 | 66.7 | 67.2 | 68.5 | 68.4 | 68.2 | 68.8 | 68.2 | 68.8
p4 | 58.3 | 58.0 | 57.9 | 58.4 | 57.9 | 58.4 | 48.8 | 48.5 | 48.3 | 48.9 | 48.3 | 48.9 | 53.0 | 52.8 | 52.7 | 53.2 | 52.7 | 53.2 | 54.5 | 54.3 | 54.1 | 54.7 | 54.1 | 54.7
p5 | 72.2 | 71.9 | 71.7 | 72.4 | 71.7 | 72.4 | 63.2 | 62.9 | 62.7 | 63.4 | 62.7 | 63.4 | 67.2 | 66.9 | 66.7 | 67.3 | 66.7 | 67.3 | 68.7 | 68.5 | 68.2 | 69.0 | 68.2 | 69.0
p6 | 67.7 | 67.5 | 67.4 | 67.9 | 67.4 | 67.9 | 58.2 | 58.1 | 57.9 | 58.4 | 57.9 | 58.4 | 62.4 | 62.3 | 62.1 | 62.6 | 62.1 | 62.6 | 64.0 | 63.9 | 63.6 | 64.2 | 63.6 | 64.2
p7 | 76.6 | 76.4 | 76.2 | 76.8 | 76.2 | 76.8 | 68.4 | 68.2 | 68.0 | 68.6 | 68.0 | 68.6 | 72.0 | 71.7 | 71.6 | 72.1 | 71.6 | 72.1 | 73.5 | 73.3 | 73.1 | 73.7 | 73.1 | 73.7
p8 | 67.3 | 66.9 | 66.7 | 67.4 | 66.7 | 67.4 | 57.9 | 57.6 | 57.4 | 58.1 | 57.4 | 58.1 | 62.1 | 61.7 | 61.5 | 62.2 | 61.5 | 62.2 | 63.6 | 63.3 | 63.0 | 63.8 | 63.0 | 63.8
p9 | 63.2 | 63.2 | 63.0 | 63.4 | 63.0 | 63.4 | 53.6 | 53.6 | 53.4 | 53.8 | 53.4 | 53.8 | 57.9 | 57.9 | 57.7 | 58.1 | 57.7 | 58.1 | 59.4 | 59.5 | 59.2 | 59.7 | 59.2 | 59.7
p10 | 73.3 | 73.2 | 73.0 | 73.5 | 73.0 | 73.5 | 64.6 | 64.5 | 64.3 | 64.8 | 64.3 | 64.8 | 68.5 | 68.3 | 68.2 | 68.6 | 68.2 | 68.6 | 70.0 | 69.9 | 69.6 | 70.2 | 69.6 | 70.2
p11 | 68.6 | 68.4 | 68.3 | 68.8 | 68.3 | 68.8 | 59.6 | 59.4 | 59.3 | 59.8 | 59.3 | 59.8 | 63.6 | 63.4 | 63.3 | 63.8 | 63.3 | 63.8 | 65.1 | 65.0 | 64.8 | 65.3 | 64.8 | 65.3
p12 | 67.5 | 67.3 | 67.1 | 67.7 | 67.1 | 67.7 | 58.2 | 58.0 | 57.8 | 58.4 | 57.8 | 58.4 | 62.3 | 62.1 | 62.0 | 62.5 | 62.0 | 62.5 | 63.8 | 63.7 | 63.5 | 64.1 | 63.5 | 64.1
p13 | 68.3 | 68.3 | 68.1 | 68.5 | 68.1 | 68.5 | 58.8 | 58.9 | 58.6 | 59.0 | 58.6 | 59.0 | 63.0 | 63.0 | 62.9 | 63.2 | 62.9 | 63.2 | 64.6 | 64.7 | 64.4 | 64.8 | 64.4 | 64.8
p14 | 67.4 | 67.2 | 67.1 | 67.6 | 67.1 | 67.6 | 58.3 | 58.1 | 57.9 | 58.4 | 57.9 | 58.4 | 62.3 | 62.1 | 62.0 | 62.5 | 62.0 | 62.5 | 63.8 | 63.7 | 63.5 | 64.1 | 63.5 | 64.1
p15 | 61.9 | 61.5 | 61.4 | 62.0 | 61.4 | 62.0 | 52.1 | 51.8 | 51.6 | 52.3 | 51.6 | 52.3 | 56.4 | 56.2 | 56.0 | 56.6 | 56.0 | 56.6 | 58.0 | 57.7 | 57.5 | 58.2 | 57.5 | 58.2
p16 | 68.8 | 68.7 | 68.5 | 68.9 | 68.5 | 68.9 | 59.6 | 59.5 | 59.3 | 59.8 | 59.3 | 59.8 | 63.6 | 63.5 | 63.4 | 63.8 | 63.4 | 63.8 | 65.2 | 65.1 | 64.9 | 65.4 | 64.9 | 65.4
| p13 (Wade) | p14 (White) | p15 (Whitlock) | p16 (Wright)
E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N | E-E | N-N | N-E | E-N | N-B | B-N
p1 | 66.6 | 66.5 | 66.4 | 66.7 | 66.3 | 66.7 | 66.7 | 66.7 | 66.4 | 66.9 | 66.4 | 66.9 | 71.5 | 71.5 | 71.3 | 71.7 | 71.3 | 71.7 | 65.8 | 65.7 | 65.5 | 65.9 | 65.5 | 65.9
p2 | 66.4 | 66.1 | 66.0 | 66.6 | 66.0 | 66.6 | 66.5 | 66.3 | 66.1 | 66.7 | 66.1 | 66.7 | 71.4 | 71.3 | 71.0 | 71.7 | 71.0 | 71.7 | 65.6 | 65.4 | 65.2 | 65.8 | 65.2 | 65.8
p3 | 68.3 | 68.0 | 67.9 | 68.4 | 67.9 | 68.4 | 68.3 | 68.2 | 68.0 | 68.5 | 68.0 | 68.5 | 73.0 | 73.0 | 72.7 | 73.3 | 72.7 | 73.3 | 67.5 | 67.3 | 67.1 | 67.6 | 67.1 | 67.6
p4 | 54.0 | 53.7 | 53.6 | 54.1 | 53.6 | 54.1 | 54.3 | 54.2 | 53.9 | 54.5 | 53.9 | 54.5 | 59.5 | 59.4 | 59.2 | 59.8 | 59.2 | 59.8 | 53.3 | 53.0 | 52.9 | 53.4 | 52.9 | 53.4
p5 | 68.5 | 68.1 | 68.0 | 68.6 | 68.0 | 68.6 | 68.5 | 68.2 | 68.0 | 68.7 | 68.0 | 68.7 | 73.4 | 73.2 | 72.9 | 73.6 | 72.9 | 73.6 | 67.6 | 67.3 | 67.1 | 67.8 | 67.1 | 67.8
p6 | 63.7 | 63.4 | 63.3 | 63.8 | 63.3 | 63.8 | 63.8 | 63.7 | 63.4 | 64.0 | 63.4 | 64.0 | 68.9 | 68.9 | 68.6 | 69.3 | 68.6 | 69.3 | 62.8 | 62.6 | 62.5 | 63.0 | 62.5 | 63.0
p7 | 73.3 | 73.0 | 72.9 | 73.4 | 72.9 | 73.4 | 73.3 | 73.0 | 72.8 | 73.5 | 72.8 | 73.5 | 77.6 | 77.5 | 77.2 | 77.9 | 77.2 | 77.9 | 72.5 | 72.2 | 72.1 | 72.7 | 72.1 | 72.7
p8 | 63.3 | 62.8 | 62.7 | 63.4 | 62.7 | 63.4 | 63.4 | 63.1 | 62.9 | 63.6 | 62.9 | 63.6 | 68.5 | 68.2 | 67.9 | 68.8 | 67.9 | 68.8 | 62.4 | 62.0 | 61.9 | 62.6 | 61.9 | 62.6
p9 | 59.0 | 58.9 | 58.8 | 59.2 | 58.8 | 59.2 | 59.2 | 59.3 | 59.1 | 59.5 | 59.1 | 59.5 | 64.5 | 64.6 | 64.3 | 64.8 | 64.3 | 64.8 | 58.2 | 58.2 | 58.0 | 58.4 | 58.0 | 58.4
p10 | 69.7 | 69.5 | 69.4 | 69.8 | 69.4 | 69.8 | 69.7 | 69.7 | 69.4 | 70.0 | 69.4 | 70.0 | 74.4 | 74.4 | 74.1 | 74.7 | 74.1 | 74.7 | 68.9 | 68.7 | 68.6 | 69.1 | 68.6 | 69.1
p11 | 64.8 | 64.5 | 64.5 | 64.9 | 64.5 | 64.9 | 64.9 | 64.8 | 64.6 | 65.1 | 64.6 | 65.1 | 69.8 | 69.7 | 69.5 | 70.1 | 69.5 | 70.1 | 64.0 | 63.8 | 63.7 | 64.2 | 63.7 | 64.2
p12 | 63.5 | 63.2 | 63.1 | 63.7 | 63.1 | 63.7 | 63.6 | 63.5 | 63.3 | 63.9 | 63.3 | 63.9 | 68.7 | 68.6 | 68.4 | 69.0 | 68.4 | 69.0 | 62.7 | 62.5 | 62.3 | 62.9 | 62.3 | 62.9
p13 | 64.3 | 64.2 | 64.1 | 64.4 | 64.1 | 64.4 | 64.4 | 64.4 | 64.2 | 64.6 | 64.2 | 64.6 | 69.5 | 69.7 | 69.4 | 69.9 | 69.4 | 69.9 | 63.4 | 63.4 | 63.2 | 63.6 | 63.2 | 63.6
p14 | 63.5 | 63.2 | 63.1 | 63.6 | 63.1 | 63.6 | 63.6 | 63.5 | 63.3 | 63.9 | 63.3 | 63.9 | 68.6 | 68.5 | 68.3 | 68.9 | 68.3 | 68.9 | 62.7 | 62.5 | 62.3 | 62.9 | 62.3 | 62.9
p15 | 57.6 | 57.2 | 57.1 | 57.7 | 57.1 | 57.7 | 57.8 | 57.6 | 57.3 | 58.0 | 57.3 | 58.0 | 63.2 | 63.0 | 62.7 | 63.5 | 62.7 | 63.5 | 56.7 | 56.4 | 56.3 | 56.9 | 56.3 | 56.9
p16 | 64.9 | 64.7 | 64.6 | 65.0 | 64.6 | 65.0 | 65.0 | 64.9 | 64.7 | 65.2 | 64.7 | 65.2 | 70.0 | 70.0 | 69.7 | 70.3 | 69.7 | 70.3 | 64.1 | 63.9 | 63.8 | 64.2 | 63.8 | 64.2

Note: Numbers are in percentages and the player from the left column throws
first while the player (name in parentheses) from the top row throws second. N
denotes the NS strategy, E denotes the equilibrium strategy, and B denotes the
best-response strategy to an opponent’s NS strategy. We note that, to 1
decimal place, N-B and B-N are indistinguishable from N-E and E-N,
respectively, in the vast majority of cases.

#### Remark on Chisnall vs. Cross

Near the end of Section 6.3 we explained why the function
$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N)$ typically monotonically increases in $N$ until some value $N^{*}$
say, after which it decreases monotonically to zero. An exception to this
general rule occurs when playing $\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}$
makes A an underdog but switching to playing $\pi_{\mbox{\tiny A}}^{*}$ makes
him a favorite. In this latter case we would see $\mbox{Gain}(\pi_{\mbox{\tiny
A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ monotonically increases to
1 as $N\to\infty$ and we observed this in the case of Chisnall (player A) vs.
Cross (player B) who are very evenly matched. It may be seen from Table 2 that
when Chisnall and Cross both play their equilibrium strategies then Chisnall
has a slight edge as he wins 64.9% of the legs he starts whereas Cross wins
64.5% of the legs he starts. In contrast, if Chisnall plays his NS strategy
then Cross will win 64.6% of the legs he starts whereas Chisnall will win
64.5% of the legs he starts thereby giving Cross an edge. This then is a case
where (at least in the limit of large $N$) taking account of an opponent’s
score and playing the equilibrium strategy makes all the difference between
winning and losing.

##  Online Appendix C Computing Match-Win Probabilities

In this online appendix we describe the calculations that allow us to convert
the win-probabilities for single legs into the win-probability for a match
consisting of several legs. Following on from Section 6.3, we wish to compute
$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N)$ as defined in (7) and which we repeat here for the sake of
convenience:

$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N):=P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{*},N)-P_{\mbox{\tiny
A}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N).$ (C-11)

$\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}},N)$ is therefore player A’s increase in probability of winning a match of
$N$ legs from playing $\pi_{\mbox{\tiny A}}^{*}$ rather than $\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}}$ against player B playing $\pi_{\mbox{\tiny B}}^{*}$.
Each term in (C-11) is easily calculated. Specifically, let $V$ denote the
number of legs that A wins out of $N$ legs in total where A starts the first
leg and the players then alternate in starting legs. We let $V_{\mbox{\tiny
A}}$ and $V_{\mbox{\tiny B}}$ denote the number of legs won by A that were
started by A and B, respectively, and then we have $V=V_{\mbox{\tiny
A}}+V_{\mbox{\tiny B}}$. As a typical darts match has an odd number of legs,
we will assume $N=2K+1$ so that a player needs to win at least $K+1$ legs to
win the match. Therefore, we have

$\displaystyle P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{*},N)$
$\displaystyle=$ $\displaystyle\text{P}(V\geq K+1)$ (C-12) $\displaystyle=$
$\displaystyle\sum_{j=1}^{K+1}\text{P}(V\geq K+1\mid V_{\mbox{\tiny
A}}=j)\text{P}(V_{\mbox{\tiny A}}=j)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{K+1}\text{P}(V_{\mbox{\tiny B}}\geq
K+1-j)\text{P}(V_{\mbox{\tiny A}}=j)$

which is easily calculated since $V_{\mbox{\tiny
A}}\sim\mbox{Bin}(K+1,p_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{*}))$ and
$V_{\mbox{\tiny B}}\sim\mbox{Bin}(K,p_{\mbox{\tiny B}}(\pi_{\mbox{\tiny
A}}^{*}))$ where

$\displaystyle p_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}})$ $\displaystyle:=$
$\displaystyle\text{P}(\mbox{A wins leg}\mid\mbox{A starts leg, A plays
}\pi_{\mbox{\tiny A}})$ $\displaystyle p_{\mbox{\tiny B}}(\pi_{\mbox{\tiny
A}})$ $\displaystyle:=$ $\displaystyle\text{P}(\mbox{A wins leg}\mid\mbox{B
starts leg, A plays }\pi_{\mbox{\tiny A}}).$

$P_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ is computed as
in (C-12) except in this case $V_{\mbox{\tiny
A}}\sim\mbox{Bin}(K+1,p_{\mbox{\tiny A}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny
NS}}))$ and $V_{\mbox{\tiny B}}\sim\mbox{Bin}(K,p_{\mbox{\tiny
B}}(\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}}))$.

###  C.1 Further Results for $\mbox{Gain}(\pi_{\mbox{\tiny
A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$

Tables 4, 5 and 6 display values for $\mbox{Gain}(\pi_{\mbox{\tiny
A}}^{*},\pi_{\mbox{\tiny A}}^{\mbox{\tiny NS}},N)$ when $N=1$, $21$ and $35$,
respectively.

Table 4: $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ in a Match of $N=1$ Leg

| p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 | p11 | p12 | p13 | p14 | p15 | p16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
p1 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
p2 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
p3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.3 | 0.3 | 0.3
p4 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
p5 | 0.5 | 0.5 | 0.5 | 0.4 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
p6 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
p7 | 0.4 | 0.4 | 0.4 | 0.4 | 0.5 | 0.5 | 0.4 | 0.4 | 0.4 | 0.5 | 0.4 | 0.4 | 0.5 | 0.4 | 0.4 | 0.4
p8 | 0.5 | 0.5 | 0.5 | 0.5 | 0.6 | 0.6 | 0.5 | 0.5 | 0.5 | 0.6 | 0.5 | 0.5 | 0.6 | 0.5 | 0.5 | 0.6
p9 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
p10 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3
p11 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.4 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.3 | 0.3 | 0.3
p12 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
p13 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
p14 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
p15 | 0.5 | 0.4 | 0.5 | 0.4 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.4 | 0.5 | 0.5 | 0.4 | 0.4 | 0.5
p16 | 0.3 | 0.3 | 0.3 | 0.2 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3

Note: Numbers are in percentages. The player on the left column is player A
who begins the first leg of the match. The player on the top row is player B
who uses his equilibrium strategy $\pi_{\mbox{\tiny B}}^{*}$ throughout.

Table 5: $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ in a Match of $N=21$ Legs

| p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 | p11 | p12 | p13 | p14 | p15 | p16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
p1 | 0.8 | 0.8 | 0.8 | 0.4 | 0.8 | 0.8 | 0.7 | 0.7 | 0.6 | 0.8 | 0.7 | 0.7 | 0.8 | 0.7 | 0.5 | 0.8
p2 | 1.2 | 1.1 | 1.1 | 0.6 | 1.2 | 1.2 | 1.0 | 1.1 | 1.0 | 1.2 | 1.1 | 1.1 | 1.3 | 1.1 | 0.9 | 1.2
p3 | 0.9 | 0.9 | 0.9 | 0.4 | 1.0 | 0.9 | 0.8 | 0.9 | 0.7 | 1.0 | 0.8 | 0.9 | 1.0 | 0.8 | 0.6 | 0.9
p4 | 0.8 | 0.8 | 0.7 | 1.1 | 0.8 | 1.1 | 0.4 | 1.1 | 1.3 | 0.6 | 0.9 | 1.0 | 1.1 | 1.0 | 1.3 | 1.0
p5 | 1.5 | 1.5 | 1.6 | 0.7 | 1.6 | 1.5 | 1.4 | 1.4 | 1.1 | 1.6 | 1.4 | 1.4 | 1.6 | 1.4 | 1.0 | 1.5
p6 | 1.3 | 1.3 | 1.2 | 0.8 | 1.3 | 1.4 | 1.0 | 1.3 | 1.2 | 1.2 | 1.2 | 1.3 | 1.4 | 1.3 | 1.1 | 1.4
p7 | 1.0 | 1.0 | 1.1 | 0.3 | 1.2 | 0.9 | 1.2 | 0.8 | 0.6 | 1.2 | 0.9 | 0.9 | 1.0 | 0.8 | 0.5 | 1.0
p8 | 1.5 | 1.5 | 1.5 | 1.1 | 1.5 | 1.7 | 1.0 | 1.6 | 1.6 | 1.4 | 1.5 | 1.7 | 1.8 | 1.6 | 1.5 | 1.7
p9 | 0.6 | 0.6 | 0.5 | 0.5 | 0.5 | 0.7 | 0.3 | 0.6 | 0.7 | 0.5 | 0.6 | 0.6 | 0.7 | 0.6 | 0.7 | 0.6
p10 | 0.9 | 0.9 | 1.0 | 0.3 | 1.0 | 0.8 | 0.9 | 0.8 | 0.6 | 1.0 | 0.8 | 0.8 | 0.9 | 0.8 | 0.5 | 0.9
p11 | 0.9 | 0.9 | 0.9 | 0.6 | 1.0 | 1.0 | 0.7 | 0.9 | 0.9 | 0.9 | 0.9 | 1.0 | 1.1 | 0.9 | 0.8 | 1.0
p12 | 1.1 | 1.1 | 1.1 | 0.8 | 1.2 | 1.2 | 0.8 | 1.2 | 1.1 | 1.1 | 1.1 | 1.2 | 1.3 | 1.1 | 1.0 | 1.2
p13 | 0.6 | 0.6 | 0.5 | 0.4 | 0.6 | 0.6 | 0.4 | 0.6 | 0.5 | 0.5 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 0.6
p14 | 1.1 | 1.1 | 1.1 | 0.8 | 1.2 | 1.2 | 0.8 | 1.2 | 1.1 | 1.1 | 1.1 | 1.2 | 1.3 | 1.1 | 1.0 | 1.2
p15 | 1.1 | 1.1 | 1.0 | 1.2 | 1.0 | 1.4 | 0.6 | 1.3 | 1.5 | 0.9 | 1.1 | 1.3 | 1.4 | 1.2 | 1.5 | 1.2
p16 | 0.8 | 0.8 | 0.8 | 0.5 | 0.9 | 0.9 | 0.6 | 0.8 | 0.8 | 0.8 | 0.8 | 0.9 | 1.0 | 0.8 | 0.7 | 0.9

Note: Numbers are in percentages. The player on the left column is player A
who begins the first leg of the match. The player on the top row is player B
who uses his equilibrium strategy $\pi_{\mbox{\tiny B}}^{*}$ throughout.

Table 6: $\mbox{Gain}(\pi_{\mbox{\tiny A}}^{*},\pi_{\mbox{\tiny
A}}^{\mbox{\tiny NS}},N)$ in a Match of $N=35$ Legs

| p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 | p11 | p12 | p13 | p14 | p15 | p16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
p1 | 1.0 | 1.0 | 1.0 | 0.3 | 1.1 | 1.0 | 0.8 | 0.9 | 0.7 | 1.0 | 0.9 | 0.9 | 1.0 | 0.9 | 0.6 | 1.0
p2 | 1.5 | 1.5 | 1.4 | 0.5 | 1.5 | 1.5 | 1.1 | 1.4 | 1.1 | 1.4 | 1.4 | 1.4 | 1.6 | 1.4 | 0.9 | 1.5
p3 | 1.2 | 1.2 | 1.2 | 0.3 | 1.3 | 1.1 | 1.0 | 1.0 | 0.7 | 1.2 | 1.1 | 1.1 | 1.2 | 1.0 | 0.6 | 1.1
p4 | 0.7 | 0.7 | 0.6 | 1.5 | 0.6 | 1.1 | 0.2 | 1.1 | 1.5 | 0.5 | 0.9 | 1.1 | 1.1 | 1.0 | 1.6 | 0.9
p5 | 1.9 | 1.9 | 2.0 | 0.5 | 2.1 | 1.8 | 1.6 | 1.7 | 1.2 | 2.0 | 1.7 | 1.7 | 1.9 | 1.7 | 1.0 | 1.9
p6 | 1.6 | 1.6 | 1.5 | 0.8 | 1.6 | 1.8 | 0.9 | 1.7 | 1.5 | 1.4 | 1.6 | 1.7 | 1.9 | 1.6 | 1.3 | 1.7
p7 | 1.2 | 1.1 | 1.3 | 0.2 | 1.4 | 0.9 | 1.5 | 0.8 | 0.4 | 1.5 | 0.9 | 0.8 | 1.0 | 0.8 | 0.3 | 1.0
p8 | 1.9 | 1.9 | 1.7 | 1.2 | 1.8 | 2.2 | 1.0 | 2.1 | 2.0 | 1.6 | 1.9 | 2.1 | 2.3 | 2.0 | 1.8 | 2.1
p9 | 0.6 | 0.6 | 0.5 | 0.6 | 0.6 | 0.8 | 0.2 | 0.8 | 0.9 | 0.5 | 0.7 | 0.8 | 0.8 | 0.7 | 0.8 | 0.7
p10 | 1.2 | 1.1 | 1.2 | 0.2 | 1.3 | 1.0 | 1.1 | 0.9 | 0.6 | 1.3 | 1.0 | 1.0 | 1.1 | 0.9 | 0.5 | 1.1
p11 | 1.2 | 1.2 | 1.1 | 0.6 | 1.2 | 1.3 | 0.7 | 1.2 | 1.0 | 1.1 | 1.1 | 1.2 | 1.4 | 1.2 | 0.9 | 1.3
p12 | 1.4 | 1.4 | 1.3 | 0.8 | 1.4 | 1.6 | 0.8 | 1.5 | 1.4 | 1.2 | 1.4 | 1.5 | 1.7 | 1.5 | 1.2 | 1.5
p13 | 0.7 | 0.7 | 0.6 | 0.4 | 0.7 | 0.8 | 0.4 | 0.7 | 0.6 | 0.6 | 0.7 | 0.7 | 0.8 | 0.7 | 0.6 | 0.8
p14 | 1.4 | 1.4 | 1.3 | 0.8 | 1.4 | 1.6 | 0.8 | 1.5 | 1.4 | 1.2 | 1.4 | 1.5 | 1.7 | 1.5 | 1.2 | 1.5
p15 | 1.1 | 1.1 | 0.9 | 1.5 | 1.0 | 1.6 | 0.4 | 1.5 | 1.9 | 0.8 | 1.2 | 1.5 | 1.6 | 1.4 | 1.9 | 1.4
p16 | 1.1 | 1.1 | 1.0 | 0.5 | 1.1 | 1.1 | 0.7 | 1.1 | 0.9 | 1.0 | 1.0 | 1.1 | 1.2 | 1.0 | 0.8 | 1.1

Note: Numbers are in percentages. The player on the left column is player A
who begins the first leg of the match. The player on the top row is player B
who uses his equilibrium strategy $\pi_{\mbox{\tiny B}}^{*}$ throughout.

##  Online Appendix D Other Problem Formulations

In this online appendix we describe some additional problem formulations
including a formulation where we model the hot-hands phenomenon within a turn.
In Online Appendix D.1 we consider the non-strategic DP formulation when we
ignore the turn feature. In Online Appendix D.2 we briefly explain how we
could easily account for the hot-hands phenomenon within a turn.

###  D.1 Ignoring the “Turn” Feature in the DP Formulation

An interesting problem in its own right is the problem whereby a player aims
to minimize the expected number of throws to get to zero. Such a player pays
no attention to the score of his opponent and is therefore non-strategic. To
the best of our knowledge Kohler (1982) was the first to solve this problem.
(He used the skill model $(x,y)\sim\mbox{N}_{2}({\bf a},\sigma^{2}{\bf I})$
and due to limited computing power at the time, used various numerical
approximations together with a branch-and-bound technique to solve the DP.)

The state space for the DP is $(s,i)$ where $s\in\\{0,2,\ldots,501\\}$ denotes
the player’s current score and $i$ denotes how many throws have already been
taken. The state transition function $f(s_{i},z)$ depends on the current state
$s_{i}$ and the realized dart score. It is defined as follows.

$s_{i+1}=f(s_{i},z):=\left\\{\begin{array}[]{ll}s_{i}-h(z),&\ s_{i}-h(z)>1\\\
0,&\ h(z)=s_{i}\mbox{ and }z\mbox{ a double}\\\ s_{i},&\
\hbox{otherwise}\end{array}\right.$ (D-13)

where $h(z)\geq 0$ is the numerical score when the $(i+1)^{st}$ throw achieves
score $z$. The three cases in (D-13) correspond to a valid throw, checking
out, and going bust, respectively. Let $V_{\mbox{\tiny NS}}(s,i)$ denote the
value function, and then the Bellman equation satisfies

$\displaystyle V_{\mbox{\tiny NS}}(s_{i},i)$ $\displaystyle=$
$\displaystyle\min_{{\bf a}\in{\cal
A}}\text{E}\left[V(s_{i+1},i+1)\mid(s_{i},i),{\bf a}\right]$ (D-14)
$\displaystyle=$ $\displaystyle\min_{{\bf a}\in{\cal
A}}\,\sum_{z}\,V_{\mbox{\tiny NS}}(f(s_{i},z),i+1)\,p(z;\,{\bf a})$

with the terminal condition $V_{\mbox{\tiny NS}}(0,i)=i$. In this non-
strategic version of the game we want to compute $V_{\mbox{\tiny NS}}(501,0)$.

As with the DP in Appendix B.1, there is a monotonic structure to this DP.
Specifically, the state $s_{i}$ is non-increasing in $i$ so that $s_{i+1}\leq
s_{i}$ for all $i$. We can take advantage of this by redefining
$V_{\mbox{\tiny NS}}(s)$ to be the minimum expected number of throws required
to reach zero starting from state $s\geq 2$. We then have

$\displaystyle V_{\mbox{\tiny NS}}(s)$ $\displaystyle=$
$\displaystyle\min_{{\bf a}\in{\cal A}}\left\\{1+\text{E}[V_{\mbox{\tiny
NS}}(f(s,z))\mid s,{\bf a}]\right\\}$ (D-15) $\displaystyle=$
$\displaystyle\min_{{\bf a}\in{\cal A}}\left\\{1+\sum_{z}V_{\mbox{\tiny
NS}}(f(s,z))\,p(z;\,{\bf a})\right\\}$

with $V_{\mbox{\tiny NS}}(0):=0$. We can solve (D-15) successively for
$s=2,\ldots,501$.

###  D.2 Modeling a Within-Turn “Hot-Hand” Phenomenon

Given a sufficiently rich data-set, one could easily estimate a different
skill model $p_{i}(x,y;{\bf a})$ for each throw $i$ within a turn. Indeed it
is well known, see e.g. Ötting et al. (2020), that the success rates for the
second and third throws within a turn are considerably higher than the success
rate for the first throw. An obvious and widely accepted explanation for this
is that errors in the first throw can be used to re-calibrate and hence
improve the accuracy of the second and third throws.

Ötting et al. (2020) also provide some evidence for the hot-hands phenomenon
within a turn but find little evidence for it across turns. Here we simply
observe that it would be straightforward to model the within-turn hot-hands
phenomenon by a simple and small expansion of the state space. Consider the DP
formulation in Appendix B.1, for example. Recall the state space for this DP
consisted of $(s,i,u)$ where $s$ is the score at the beginning of the turn,
$i\in\\{1,2,3\\}$ is the number of throws remaining in the turn and $u$ is the
score-to-date within the turn. To model the hot-hands phenomenon within a
turn, we instead use state vectors $(s,i,u)$ where $s$ and $u$ are as before
but $i\in\\{1,\ldots,7\\}$ used to encode the number of throws remaining in
the turn (as before) as well as the success(es) of any previous throw(s)
within the turn. For example, we could use $i=1$ to represent that there are 3
throws remaining in the turn. Then $i=2$ and $i=3$ could denote the states
where there are 2 remaining throws in the turn and the first throw was a
success or failure, respectively. Finally, we could use $i=4,\ldots,7$ to
denote the states where there is just 1 throw remaining. There are 4 such
states depending on the success or failure outcomes of the first 2 throws.

This formulation requires slightly more than doubling the number of states and
it clearly also applies to the BR and ZSG problem formulations. It would
therefore only require a modest increase in computational work to solve for
these problems. The main difficulty of course would be in obtaining a
sufficiently rich data-set to estimate the probability distributions required
for the state transitions.

##  Online Appendix E Speeding Up Solving the ZSG

As noted at the beginning of Section 6.2, we were able to obtain significant
improvements in the performance of Algorithm 2 through the use of
vectorization that takes advantage of GPUs. Beyond standard uses of
vectorization (e.g. evaluating all actions for a state in a vectorized
operation, which we also implemented), we found the following use of
vectorization to be particularly efficient.

The core step of Algorithm 2 is in applying Algorithm 1 for solving the best-
response problem in the while loop. Consider then this step where we implement
Algorithm 1 using policy iteration for a given $(s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}})$. We need to update $J^{\mbox{\tiny
A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)$ via (6) for all $(i,u)$ in
the policy improvement step. It is clear from the state transition dynamics in
(4) that the value of $i$ must change during each state transition, i.e., the
state $(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)$ transitions to either the
next throw, the next turn, going bust, or the end of the game, all of which
result in a change in the value of $i$. Therefore, updating $J^{\mbox{\tiny
A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i,u)$ in the policy improvement
step does not require $J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny
B}},i,u^{\prime})$ for any states $(s_{\mbox{\tiny A}},s_{\mbox{\tiny
B}},i,u^{\prime})$. This means that $J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}},i,u)$ can be updated simultaneously for states with the
same value of $i$. Specifically, given $(s_{\mbox{\tiny A}},s_{\mbox{\tiny
B}})$, we update $J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny
B}},i=1,u)$ for all $u$ in one operation, then update $J^{\mbox{\tiny
A-BR}}(s_{\mbox{\tiny A}},s_{\mbox{\tiny B}},i=2,u)$ for all $u$ in another
operation, and finally update $J^{\mbox{\tiny A-BR}}(s_{\mbox{\tiny
A}},s_{\mbox{\tiny B}},i=3,u)$. To achieve this, we load all the information
required into several large matrices and then evaluate the Bellman equation
(6) using a GPU. GPUs of course are extremely efficient in executing large
matrix algebra.