Confidence Intervals for the Shapley–Shubik Power Index in Markovian Games

Abstract

We consider simple Markovian games, in which several states succeed each other over time, following an exogenous discrete-time Markov chain. In each state, a different simple static game is played by the same set of players. We investigate the approximation of the Shapley–Shubik power index in simple Markovian games (SSM). We prove that an exponential number of queries on coalition values is necessary for any deterministic algorithm even to approximate SSM with polynomial accuracy. Motivated by this, we propose and study three randomized approaches to compute a confidence interval for SSM. They rest upon two different assumptions, static and dynamic, about the process through which the estimator agent learns the coalition values. Such approaches can also be utilized to compute confidence intervals for the Shapley value in any Markovian game. The proposed methods require a number of queries, which is polynomial in the number of players in order to achieve a polynomial accuracy.

Appendices

Appendix A: Proof of Theorem 2

Proof

We will prove that there exists a class $\mathcal{F}$ of game instances for which any deterministic algorithm computing $\mathrm {SS}_{j}^{(s)}$ with accuracy of at least 1/(2P) must utilize $\varOmega(2^{P}/\sqrt{P})$ queries. Similarly to [8], let us construct $\mathcal{F}$ when P is odd. Let $\varLambda\subseteq\mathcal{P}\backslash\{j\}$. There exists a set D _o of $\binom{P-1}{[P-1]/2}/2$ coalitions of cardinality [P−1]/2 such that player j is critical only for D _o . In particular, for |Λ|≤[P−1]/2, v ^(s)(Λ)=0; if |Λ|=[P−1]/2, then, if Λ∈D _o , v ^(s)(Λ∪{j})=1, otherwise v ^(s)(Λ∪{j})=0. The values of the remaining coalitions are 1 if and only if they contain a winning coalition among the ones constructed so far. The Shapley value for player j is thus

$$\mathrm {SS}_j^{(s)} = \frac{([P-1]/2)! ([P-1]/2)!}{2(P)!} \binom {P-1}{[P-1]/2} = \frac{1}{2P}. $$

Hence, for any deterministic algorithm ALG _o employing a number of queries smaller than μ _o (P), where

$$\mu_o(P) = \frac{1}{2}\binom{P-1}{[P-1]/2}, $$

there always exists an instance belonging to $\mathcal{F}$ for which ALG _o would answer $\mathrm {SS}_{j}^{(s)}=0$. By Stirling’s approximation, we can say that $\mu_{o}(P) \in\varOmega(2^{P}/\sqrt{P})$. Let us now construct the class $\mathcal{F}$ of instances when P is even and P>2. Let D _e be a set of $\binom{P-2}{[P-2]/2}$ coalitions of cardinality [P−2]/2, belonging to $\mathcal{C}\backslash\{j\}$, such that player j is critical only for D _e . Then

$$\mathrm {SS}_j^{(s)} = \frac{(P/2-1)! (P/2)!}{(P)!} \binom{P-2}{[P-2]/2} = \frac{1}{2[P-1]}>\frac{1}{2P}. $$

Similarly to before, for any deterministic algorithm ALG _e using a number of queries smaller than

$$\mu_e(P) = \binom{P-1}{[P-2]/2} -\binom{P-2}{[P-2]/2} = \frac{P-2}{P} \binom{P-2}{[P-2]/2}, $$

there always exists an instance belonging to $\mathcal{F}$ for which ALG _e would answer $\mathrm {SS}_{j}^{(s)}=0$. By Stirling approximation, we can say that $\mu_{e}(P)\in\varOmega(2^{P}/\sqrt{P})$. Hence, a number of samples $\mu\in\varOmega(2^{P}/\sqrt{P})$ is needed to achieve an accuracy of at least 1/(2P). Hence, the thesis is proved. □

Appendix B: Proof of Corollary 1

Proof

Any deterministic algorithm employs a certain number of queries in each state s in order to compute $\mathrm {SSM}_{j}(\varGamma_{s})=\sum_{i=1}^{|S|} \boldsymbol{\sigma}_{i}(s)\mathrm {SS}_{j}^{(s_{i})}$. Let I ₀ be a game instance in which player j is a dummy player in all the single stage games {v ^(s)} _s∈S , i.e., $\mathrm {SS}_{j}^{(s)}=0$ for all s∈S. Let I ₁ be a game instance such that $\mathrm {SS}_{j}^{(s)}=0$ for all s except for s _k , for which σ(s _k )≠0, and such that the game $\varPsi ^{(s_{k})}$ belongs to the class $\mathcal{F}$ of instances described in the proof of Theorem 2. Therefore,

$$\mathrm {SSM}_j(\varGamma_s)= \frac{\boldsymbol{\sigma}_k(s)}{2P} $$

in the case that P is odd and

$$\mathrm {SSM}_j(\varGamma_s)= \frac{\boldsymbol{\sigma}_k(s)}{2[P-1]} $$

if P is even. Hence, any deterministic algorithm needs $\varOmega (2^{P}/\sqrt{P})$ queries in state s _k to achieve an accuracy better than σ _k (s)/(2P). Set c=σ _k (s)/2. Hence, the thesis is proved. □

Appendix C: Proof of Lemma 1

Proof

We will provide the proof for continuous random variables; the proof for the discrete case is totally similar. By induction, it is sufficient to prove that, if Pr(A ₁∈[l ₁;r ₁])≥1−δ ₁ and Pr(A ₂∈[l ₂;r ₂])≥1−δ ₂, then

$$\Pr \bigl(A_1+A_2\in[l_1+l_2 ; r_1;r_2] \bigr) \ge (1-\delta _1) (1- \delta_2). $$

Let f _A be the probability density function of the r.v. A. Let , i=1,2. Then

Hence, the thesis is proved. □

Appendix D: Proof of Theorem 4

Proof

Let us consider the following constrained minimization problem over the reals:

$$ \left\{ \begin{array}{@{}l@{}} \min_{ \omega_1,\ldots,\omega_{|S|}} \sum _{i=1}^{|S|} \boldsymbol{\sigma}_i^2(s)/ \omega_i , \\ \noalign{\vspace*{6pt}} \sum_{i=1}^{|S|} \omega_i = n,\quad \omega_i\in\mathbb{R}. \end{array}\right. $$

(12)

By using, e.g., the Lagrangian multiplier technique, it is easy to see that the optimum value for ω _i is

$$\omega_i^* = \frac{\boldsymbol{\sigma}_i(s) n}{\sum_{k=1}^{|S|}\boldsymbol{\sigma}_k(s)} $$

and that the minimum value of the objective function is

$$ \xi^* = \frac{ [ \sum_{i=1}^{|S|} \boldsymbol{\sigma}_i(s) ]^2}{n}. $$

(13)

The value ξ ^∗ clearly represents a lower bound for the optimization problem over the integers in the case of simple games. Since we deal with the average criterion, let σ _i (s)≡π _i . Now we can find a lower bound for $\sqrt{n} \widetilde{\varepsilon}(n,\delta)$ over n that does not depend on the number of queries n:

(14)

and the optimum value of x _i in (14) is

$$x^*_i = \frac{\boldsymbol{\pi}_i}{\sum_{k=1}^{|S|} \boldsymbol{\pi }_k} = \boldsymbol{\pi}_i . $$

For Theorem 3,

$$n_i/n \stackrel{n\uparrow\infty}{\longrightarrow} \boldsymbol{\pi }_i \quad\mathrm{with\ probability\ 1}. $$

Hence, n _i /n converges with probability 1 to the optimum value $x^{*}_{i}$ and, by continuity, the thesis is proved. □

Appendix E: Proof of Lemma 2

Proof

In the case of simple Markovian games, the optimization problem (9) turns into

$$ \left\{ \begin{array}{@{}l@{}} \min_{n_1, \ldots,n_{|S|}} \sum_{i=1}^{|S|} \boldsymbol{\sigma }_i^2(s)/n_i, \\ \noalign{\vspace*{6pt}} \sum_{i=1}^{|S|} n_i = n, \qquad n_i\in\mathbb{N}. \end{array}\right. $$

(15)

Let us consider the constrained minimization problem over the reals in (12). Since evidently ξ ^∗, defined in (13), is not greater than the minimum value of the objective function in (15), then by straightforward inspection over the expressions (5) and (8) the thesis is proved. □