# A Two-Player Resource-Sharing Game with Asymmetric Information

^{*}

## Abstract

**:**

## 1. Introduction

- We consider the problem of a two-player singleton stochastic resource-sharing game with asymmetric information. We first provide an iterative best response algorithm to find an $\u03f5$-approximate Nash equilibrium of the system. This equilibrium analysis uses potential game concepts.
- When the players do not trust each other and place no assumptions on the incentives of the opponent, we solve the problem of maximizing the worst-case expected utility of the first player using a novel algorithm that leverages techniques from online optimization and the drift-plus penalty methods. The algorithm developed can be used to solve the general unconstrained problem of finding the randomized decision $\alpha \in \{1,2,\dots ,n\}$, which maximizes $\mathbb{E}\left\{h\right(\mathit{x};\mathbf{\Theta}\left)\right\}$, where $\mathit{x}\in {\mathbb{R}}^{n}$ with ${x}_{k}=\mathbb{E}\left\{{\Gamma}_{k}{\mathbb{1}}_{\{\alpha =k\}}\right\}$, $\mathbf{\Theta}\in {\mathbb{R}}^{m}$ and $\mathbf{\Gamma}\in {\mathbb{R}}^{n}$ are non-negative random vectors with finite second moments, and h is a concave function such that $\tilde{h}\left(\mathit{x}\right)=\mathbb{E}\left\{h(\mathit{x};\mathbf{\Theta})\right\}$ is Lipschitz continuous, entry-wise non-decreasing and has bounded subgradients.
- We show our algorithm uses a mixture of only $\mathcal{O}(1/{\epsilon}^{2})$ pure strategies using a detailed analysis of the sample path of the related virtual queues (our preliminary work on this algorithm used a mixture of $\mathcal{O}(1/{\epsilon}^{3})$ pure strategies). Virtual queues are also used for constrained online convex optimization in [13], but our problem structure is different and requires a different and more involved treatment.

#### 1.1. Background on Resource-Sharing Games

#### 1.2. Notation

## 2. Materials and Methods

## 3. Formulation

## 4. Computing the $\mathit{\u03f5}$-Approximate Nash Equilibrium

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 5. Worst-Case Expected Utility

**Lemma**

**2.**

**Proof**

**of**

**Lemma**

**2.**

**Theorem**

**2.**

- 1.
- is concave.
- 2.
- is entry-wise non-decreasing.
- 3.
- satisfies,$$\begin{array}{c}\hfill |f\left(\mathit{x}\right)-f\left(\mathit{y}\right)|\le \frac{3}{2}\sum _{j\in \mathcal{A}}|{x}_{j}-{y}_{j}|+\frac{3}{2}\sum _{j\in {\mathcal{A}}^{c}}{E}_{j}|{x}_{j}-{y}_{j}|,\end{array}$$

#### 5.1. Explicit Solution for $a=b=d=0$

**Lemma**

**3.**

#### 5.2. Solving the General Case

Algorithm 1: Algorithm for the generation of the optimal mixture of T pure strategies. |

#### 5.2.1. Solving (P3)

#### 5.2.2. How Good Is the Mixed Strategy Generated by Algorithm 1

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Lemma**

**7.**

**Proof**

**of**

**Lemma**

**7.**

**Lemma**

**8.**

**Lemma**

**9.**

## 6. Simulations

- $a=0,b=0,c=3,d=0$: Both players do not have private information.
- $a=0,b=1,c=2,d=0$: Only player B has private information.
- $a=1,b=1,c=1,d=0$: Both players have private information.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Theorem 2

## Appendix B. Proof of Lemma 3

**Lemma**

**A1.**

**Proof**

**of**

**Lemma**

**A1.**

**Lemma**

**A2.**

**Proof**

**of**

**Lemma**

**A2.**

**Lemma**

**A3.**

- 1.
- ${\mu}_{k}\ge 0$ for all k such that $1\le k\le n$.
- 2.
- ${\sum}_{k=1}^{n}{\mu}_{k}=\frac{1}{2}$.
- 3.
- ${E}_{k}(1-{\mu}_{k})=\frac{r-\frac{1}{2}}{{S}_{r}}$ for $1\le k\le r$.
- 4.
- ${E}_{k}(1-{\mu}_{k})\le \frac{r-\frac{1}{2}}{{S}_{r}}$ for $r+1\le k\le n$.

**Proof**

**of**

**Lemma**

**A3.**

- Notice that by the definition of ${\mu}_{k}$, it is enough to prove the result for $1\le k\le r$. Notice that we are required to prove that$$\begin{array}{c}\hfill \frac{1}{{E}_{k}}\frac{r-\frac{1}{2}}{{S}_{r}}\le 1,\end{array}$$$$\begin{array}{c}\hfill \frac{1}{{E}_{r}}\frac{r-\frac{1}{2}}{{S}_{r}}\le 1.\end{array}$$We consider two cases.
**Case 1:**$r=1$. This case reduces to,$$\begin{array}{c}\hfill \frac{1}{2{E}_{1}}\le \frac{1}{{E}_{1}},\end{array}$$**Case 2:**$r>1$. Note that from the definition of r in (A10), we have$$\begin{array}{c}\hfill \frac{r-\frac{1}{2}}{{S}_{r}}\ge \frac{r-\frac{3}{2}}{{S}_{r-1}}.\end{array}$$After substituting ${S}_{r-1}={S}_{r}-\frac{1}{{E}_{r}}$ and rearranging, we have the desired result. - Notice that$$\begin{array}{c}\hfill \sum _{k=1}^{n}{\mu}_{k}=\sum _{k=1}^{r}{\mu}_{k}=\sum _{k=1}^{r}\left(1-\frac{1}{{E}_{k}}\frac{r-\frac{1}{2}}{{S}_{r}}\right)=r-\frac{r-\frac{1}{2}}{{S}_{r}}\sum _{k=1}^{r}\frac{1}{{E}_{k}}=r-\frac{r-\frac{1}{2}}{{S}_{r}}{S}_{r}=\frac{1}{2}.\end{array}$$
- This follows from the definition of ${\mu}_{k}$ for $1\le k\le r$.
- There is nothing to prove if $r=n$. Hence, we can assume $r<n$. Since ${\mu}_{k}=0$ for $k\ge r+1$, it suffices to prove that ${E}_{k}\le \frac{r-(1/2)}{{S}_{r}}$. Notice that if we can prove the result for $k=r+1$, we are finished since ${E}_{k}\ge {E}_{k+1}$ for $1\le k\le n$. Note that from the definition of r in (A10), we have$$\begin{array}{c}\hfill \frac{r-\frac{1}{2}}{{S}_{r}}\ge \frac{r+\frac{1}{2}}{{S}_{r+1}}.\end{array}$$After substituting ${S}_{r+1}={S}_{r}+\frac{1}{{E}_{r+1}}$ and rearranging, we have the desired result.

## Appendix C. Proof of Lemma 4

## Appendix D. Proof of Lemma 6

## Appendix E. Proof of Lemma 8

## Appendix F. Proof of Lemma 9

**Lemma**

**A4.**

- 1.
- ${Q}_{j}(t+1)\le {Q}_{j}\left(t\right)+{u}_{j}$ for all $t\ge 1$.
- 2.
- Assume ${Q}_{j}\left(t\right)\ge ({v}_{j}+\sqrt{2}{u}_{j})\sqrt{\alpha}$ for some $t\ge 1$. Then we have either ${\gamma}_{j}\left(t\right)=0$ or$$\begin{array}{c}\hfill {\gamma}_{j}\left(t\right)\le {\gamma}_{j}(t-1)-\frac{{u}_{j}}{\sqrt{2\alpha}}.\end{array}$$
- 3.
- Assume ${Q}_{j}\left(\tau \right)\ge ({v}_{j}+\sqrt{2}{u}_{j})\sqrt{\alpha}$ for all $\tau \in [t:t+{t}_{0}]$, where $t\ge 1$ and ${t}_{0}\ge 0$. Additionally assume ${\gamma}_{j}(t-1)=0$. Then ${\gamma}_{j}\left(\tau \right)=0$ for all $\tau \in [t-1:t+{t}_{0}]$.

**Proof**

**of**

**Lemma**

**A4.**

- Notice that from the definition of ${Q}_{j}(t+1)$ in (35), for $j\in \mathcal{A}$ we have$$\begin{array}{cc}\hfill {Q}_{j}(t+1)& =max\left\{{Q}_{j}\left(t\right)+{\gamma}_{j}\left(t\right)-{X}_{j}\left(t\right){\mathbb{1}}_{\{{\alpha}^{A}\left(t\right)=j\}},0\right\}\le max\left\{{Q}_{j}\left(t\right)+{u}_{j},0\right\}\hfill \\ & ={Q}_{j}\left(t\right)+{u}_{j},\hfill \end{array}$$
- Notice that if ${\gamma}_{j}\left(t\right)\ne 0$ then we have$$\begin{array}{c}\hfill {\gamma}_{j}\left(t\right)\le {\gamma}_{j}(t-1)-\frac{-V{f}_{t,j}^{{}^{\prime}}\left(\gamma (t-1)\right)+{Q}_{j}\left(t\right)}{2\alpha},\end{array}$$$$\begin{array}{cc}\hfill {\gamma}_{j}\left(t\right)& \le {\gamma}_{j}(t-1)-\frac{-V{f}_{t,j}^{{}^{\prime}}\left(\gamma (t-1)\right)+{Q}_{j}\left(t\right)}{2\alpha}\hfill \\ & {\le}_{\left(a\right)}{\gamma}_{j}(t-1)-\frac{-V{v}_{j}+({v}_{j}+\sqrt{2}{u}_{j})\sqrt{\alpha}}{2\alpha}{\le}_{\left(b\right)}{\gamma}_{j}(t-1)-\frac{{u}_{j}}{\sqrt{2\alpha}},\hfill \end{array}$$
- Notice if we prove ${\gamma}_{j}\left(t\right)=0$, we can use the same argument inductively to establish the result. Assume the contrary that ${\gamma}_{j}\left(t\right)\ne 0$. Then, from part 2, we should have$$\begin{array}{c}\hfill {\gamma}_{j}\left(t\right)\le {\gamma}_{j}(t-1)-\frac{{u}_{j}}{\sqrt{2\alpha}}=-\frac{{u}_{j}}{\sqrt{2\alpha}},\end{array}$$

**Case 1:**${Q}_{j}\left(t\right)\le ({v}_{j}+2\sqrt{2}{u}_{j})\sqrt{\alpha}$. This case follows from Lemma A4-1.

**Case 2:**$t\le \sqrt{2\alpha}+1$. Notice that

**Case 3:**$t>\sqrt{2\alpha}+1$ and ${Q}_{j}\left(t\right)>({v}_{j}+2\sqrt{2}{u}_{j})\sqrt{\alpha}$. For this, we prove that ${\gamma}_{j}\left(t\right)=0$, which establishes the claim from the definition of ${Q}_{j}(t+1)$ in (35) and the induction hypothesis.

## Notes

1 | Ideally, player B may not have information about ${q}_{j}^{A}$ and ${p}_{j}^{A}$. Hence, player B may not be able to utilize this exact strategy. Nevertheless, obtaining a better bound is impossible since we do not have any assumptions or information about player B’s strategy. For instance, if player B assumes that player A is using a particular strategy and if player B’s assumption turns out to be correct since player B knows the distributions of all ${W}_{j}$ for $1\le j\le n$, player B’s estimates of ${q}_{j}^{A}$ and ${p}_{j}^{A}$ are exact. |

2 | The same problem structure arises in the case with symmetric information between the players (case $a=b=0$ with d arbitrary). Hence, we can use the solution obtained in this section for the above case as well. |

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**Figure 4.**

**Top:**Case $a=0$, $b=0$, $c=3$, $d=0$.

**Middle:**Case $a=0$, $b=1$, $c=2$, $d=0$.

**Bottom:**Case $a=b=c=1$, $d=0$.

**Left:**The expected utility of the players at the $\u03f5$-approximate Nash equilibrium vs. ${E}_{1}$.

**Middle:**One possible solution for the probabilities of choosing different resources at the $\u03f5$-approximate Nash equilibrium for player A vs. ${E}_{1}$.

**Right:**One possible solution for the probabilities of choosing different resources at the $\u03f5$-approximate Nash equilibrium for player B vs. ${E}_{1}$.

**Figure 5.**

**Left:**Case $a=0$, $b=0$, $c=3$, $d=0$.

**Middle:**Case $a=0$, $b=1$, $c=2$, $d=0$.

**Right:**Case $a=b=c=1$, $d=0$.

**Top:**The maximum expected worst-case utility of player A and the error margin (shaded in blue) vs. ${E}_{1}$.

**Bottom:**One possible solution for the probabilities of choosing different resources for player A vs. ${E}_{1}$.

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**MDPI and ACS Style**

Wijewardena, M.; Neely, M.J.
A Two-Player Resource-Sharing Game with Asymmetric Information. *Games* **2023**, *14*, 61.
https://doi.org/10.3390/g14050061

**AMA Style**

Wijewardena M, Neely MJ.
A Two-Player Resource-Sharing Game with Asymmetric Information. *Games*. 2023; 14(5):61.
https://doi.org/10.3390/g14050061

**Chicago/Turabian Style**

Wijewardena, Mevan, and Michael J. Neely.
2023. "A Two-Player Resource-Sharing Game with Asymmetric Information" *Games* 14, no. 5: 61.
https://doi.org/10.3390/g14050061