 over all mixed (or behavioral) strategies σ of P1, and the minimum is over all mixed (or behavioral) strategies τ of P2.
where for two probabilities P and Q on a finite (or countable) set Y, ∥P−Q∥=∑y|P(y)−Q(y)| and D(P∥Q)=∑y∈YP(y)logP(y)Q(y) (where log denotes the natural logarithm and 0log0=0).
Proof of Theorem 2. Note that V(k,d) is monotonic increasing in d and k, and there is a positive constant C1>0 such that V(k,2)≥C1√k.
where Gxb stands for the more explicit Gb,xb. Note that ∥G∥=max(∥G1∥,∥G2∥).
A possible helpful interpretation of Γ is that nature chooses a pair x1∈X1 and x2∈X2, equivalently a pair of games Gx1 and Gx2, according to the product probability p1⊗p2. P1 is informed of the choice (Gx1,Gx2) of nature, but P2 is not. In each stage of the repeated game, both players select strategies for the first and for the second game, and P1 chooses in addition which one of the two games determines the stage payoff.
As a function of i=(i1,i2,b), for each fixed b=1,2, the payoff function Gxi,j does not depend on the coordinate ic for c≠b. Therefore we can replace the set I (which has 2|I1||I2| elements) of stage actions of P1 in the repeated game Γ with the disjoint union of I1 and I2.
Indeed, P1 can play bt=1 in stages t=1,…,k1 and bt=2 in stages t=k1+1,…,k1+k2, and the first coordinates i1t of it follow, in stages t=1,…,k1, an optimal strategy of P1 in Γ1k1(p1), and the second coordinates i2t of it follow, in stages t=k1+1,…,k1+k2, an optimal strategy of P1 in Γ2k2(p2).
For ℓ>2 and a sequence Γ1=⟨X1,p1,I1,J1,G1⟩,…,Γℓ=⟨Xℓ,pℓ,Iℓ,Jℓ,Gℓ⟩ of RGII-OS, we define by induction on ℓ the game Γ=⊗ℓb=1Γb by Γ=(⊗ℓ−1b=1Γb)⊗Γℓ.
Note that a stage action of P1 in Γ is a list of stage actions i1,…,iℓ (with ib∈Ib) and a number b (with 1≤b≤ℓ). However, given b, the payoff depends only on the coordinate ib of the stage actions. Therefore we can replace the stage actions of P1 in Γ with the disjoint union of the action sets Ib, and so with a set of size ∑b|Ib|.
Consider the example of the RGII-OS Γz=⟨X={0,1},(1/2,1/2),I,J,G⟩, introduced by Zamir [5, Section 3]. The set of states is X={0,1}, and players’ action sets are I={0,1} for P1, and J={0,1} for P2. The two payoff matrices are G0 and G1: G00,0=3, G00,1=−1, G10,0=2=−G10,1, and G∗i,j=−G∗1−i,j. Let vzk denote the normalized value of the k-stage repeated game Γz. Zamir [5] shows that limnvzn=0 and vzk≥C1/√k, where C1>0 is a positive constant.
For an arbitrary k and d≤2k, there is k≥k1>k/2 that is a multiple of ℓ:=[log2d] (≥(log2d)/2). As P1 can play (1/2,1/2) in the last k−k1 stages, kvk≥k1vk1, and thus kvk≥C1√k1√ℓ. Therefore, vk≥C12√k√logd.
4.1 Comments on the proof of Theorem 1.
Inequality (9), which is equivalent to Pinsker’s inequality, can obviously be proved directly.444 I wish to thank Stanislaw Kwapien for suggesting a proof that avoids the information-theoretic techniques, and communicating a simple analytical proof of the above displayed version of Pinsker’s inequality. The continuation of the proof avoids the (explicit) use of information-theoretic techniques.
We conclude that if ∑xp0(x)=1, then ∑kt=1E∥pt−pt−1∥1≤√2k√EH(p0).
4.2 The variation of a martingale of probabilities over a countable set.
It is of interest to find a necessary and sufficient condition for a distribution p on a countable set X for supk1√kV(k,p)<∞. We remark here on the sufficient conditions derived from the classical method and our method of bounding the variation of martingales of probabilities.
The classical bound of the variation of a martingale pk0 is obtained by bounding, for each fixed x∈X, the expectation variation E∥y(x)∥1, where y(x)∈Rk is the vector of martingale differences (p1(x)−p0(x),…,pk(x)−pk−1(x)) (thus ∥y(x)∥1=∑kt=1|pt(x)−pt−1(x)|), and summing over all x∈X. Assuming without loss of generality that p0 is a constant p∈Δ(X), we have (by the Cauchy–Schwartz inequality) ∥y(x)∥1≤√k∥y(x)∥2, and, therefore, by Jensen’s inequality, E∥y(x)∥1≤√k√E∥y(x)∥22, which by the martingale property is ≤√k√E((pk(x))2−(p0(x))2)≤√k√E((pk(x))−(p0(x))2)=√k√p0(x)−(p0(x))2. Therefore, if p∈Δ(X) and X is countable, the classical method yields that supk1√kV(k,p)<∞ whenever ∑x√p(x)<∞. As −qlogq=o(√q) as q→0+, the condition ∑x√p(x)<∞ implies that H(p)=−∑xp(x)logp(x)<∞. Obviously, there are probabilities p over a countable set X such that H(p)<∞ but ∑x√p(x)=∞. Therefore our bound provides a strictly sharper sufficient condition, H(p)<∞, for supk1√kV(k,p)<∞, compared to the one derived by using the classical method.
4.3 The asymptotic behavior of V(k,d).
The asymptotic behavior of V(k,d) deserves further study. [4] proves that V(k,2)/√k converges as k→∞ to √2π. It is of interest to find a corresponding limit theorem for V(k,d)/√klogd as 2k≥d→∞. The above-mentioned result of [4] together with our construction in the proof of Theorem 2 yields that the liminf of V(k,d)/√klog2d is ≥√2π as logdk+1/d→0 (namely, as logd=o(k) and d→∞).
4.4 Repeated games with incomplete information.
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