# Mixed Capability Games

We present a new class of strategic games, mixed capability games, as a foundation for studying how different player capabilities impact the dynamics and outcomes of strategic games. We analyze the impact of different player capabilities via a capability transfer function that characterizes the payoff of each player at equilibrium given capabilities for all players in the game. In this paper, we model a player's capability as the size of the strategy space available to that player. We analyze a mixed capability variant of the Gold and Mines Game recently proposed by Yang et al. and derive its capability transfer function in closed form.

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05/20/2022

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## 1 Introduction

Player capabilities can significantly impact the dynamics and outcomes of strategic games. Recently, Yang et al. [5] analyzed how different player capabilities affect the social welfare in several congestion games. The research models player strategies as programs in a domain-specific language and models the capability of each player as the size of the programs available to that player. All players in a given game have the same capability, with player capabilities varying across games but not within the same game.

We present mixed capability games as a general framework for studying games in which players have different capabilities, both within the same game and across different games. To capture how game outcomes depend on different player capabilities, we propose analyzing a capability transfer function that precisely quantifies the payoffs of individual players given the capabilities of all players in a game. Section 2 presents the concepts in our framework. Section 3 presents an analysis of a mixed capability game, the Mixed Gold and Mines Game, and derives closed-form expressions for the capability transfer function of this game.

## 2 Mixed Capability Games and Capability Transfer Function

We model the capability of each player as the size of the strategy space available to that player. We first present formal definitions for pure Nash equilibria of normal-form games [2], then extend the definitions to mixed Nash equilibria.

###### Definition 1

A mixed capability game is a tuple where:

• is the set of players.

• is the maximal capability of player .

• is the strategy space of player when they have capability . We also require that the strategy spaces of a player form a hierarchy: , i.e., a player has more strategies to choose from when they have higher capability.

• is the payoff function that computes the payoff for player given the strategies chosen by all players.

A specification of the actual capabilities of players is necessary to determine the outcome of the game.

###### Definition 2

A capability profile for a mixed capability game is a tuple of integers where . A capability profile determines the strategy spaces of the players. Player can choose strategies only from .

Given a capability profile , a strategy profile is a tuple where that specifies the strategies chosen by all players. A strategy profile is a pure Nash equilibrium if no player can improve their payoff by unilaterally changing their strategy: . The notation denotes a new strategy profile in which player plays strategy and all other players play the same strategy as in .

###### Definition 3

A capability transfer function of a mixed capability game is a function where denotes the integers between and , and is the power set of a set . The capability transfer function computes the set of player payoffs at equilibrium for a capability profile. Formally, given a capability profile , is a set such that if and only if there is a pure Nash equilibrium for which and .

The capability transfer function contains detailed information about the game’s behavior under varying player capabilities. Example 1 illustrates how to use a capability transfer function to define the higher level concept of a capability-positive game.

###### Example 1

Capability-positive games [5] are games in which all players share the same capability social welfare at equilibrium cannot decrease as players become more capable. Such games can be defined using the capability transfer function for that game. A game is capability-positive if where . Note that is the set of social welfare at equilibrium defined via the capability transfer function of this game.

We extend the definitions to games without pure Nash equilibria. We consider mixed Nash equilibria in which players act stochastically. All finite games have mixed Nash equilibria[1]. Given a capability profile , the strategy of a player is a distribution over possible actions, denoted as where . Player receives expected payoff :

 E[ui|s]=∑aj∈Ljcjui(a1,…,an)P(a1|s1)⋯P(an|sn)

A strategy profile is a mixed Nash equilibrium if no player can unilaterally change their own distribution to improve their expected payoff. In this case, the capability transfer function is defined as the set of expected payoffs of all mixed Nash equilibria given a capability profile.

###### Definition 4

The capability transfer function of a mixed capability game with mixed Nash equilibria is a function . Given a capability profile , is a set such that if and only if there is a mixed Nash equilibrium for which defines a distribution over and .

One natural question regarding mixed capability games is whether increasing the capability for one player does not make this player receive less payoff. Formally, let denote the set of payoffs of player at equilibrium, then the question is whether for each where and with . Example 2 shows that this is not necessarily true for Nash equilibria since the player with increased capability may switch to another strategy, which triggers responses of other players that ultimately reduce the payoff of the initial player. Note that in a Stackelberg game [4] where the leader announces their strategy before others simultaneously choose their responses, the capability transfer function is monotonic for the leader.

###### Example 2

Consider a two-player two-action bimatrix game. Player 1 is the row player with two possible capabilities: and . Player 2, the column player, has one capability: . Their payoff matrices are:

 u1=(1−120)u2=(2112)

When player 1 is has capability 1, they can only play the first row, and player 2 plays the first column, which gives payoffs 1 and 2 for each player respectively. Therefore, we have for the capability transfer function. When player 1 is allowed to use full capability, the only Nash equilibrium is (second row, second column), which gives . The capability transfer function decreases for player 1 even though player 1’s capability increases.

## 3 Mixed Gold and Mines Game

We derive exact expressions for the capability transfer function of an asymmetric version of the alternating ordering Gold and Mines Game, a special case of distance-bounded network congestion games originally proposed by Yang et al. [5]. We name this new game the Mixed Gold and Mines Game (MGMG). Unlike the previous Gold and Mines Game of Yang et al. [5] in which all players in the same game have the same capability, in a single Mixed Gold and Mine Game, players may have different capabilities.

MGMG is a two-player congestion game parameterized by five numbers . MGMG has resources arranged as a specific pattern; players use line segments to cover resources to receive payoffs. As in all congestion games, MGMG games always have pure Nash equilibria [3].

#### Resource Layout:

Each MGMG game has resources arranged on two lines. Each resource is either a gold site or a mine site. Each line contains gold sites and mine sites in alternating order. Resources are placed at distinct horizontal locations . For the resource at location , indicates which line it is placed on, and indicates whether it is a gold site () or a mine site ().

#### Game Objective:

Two players maximize their payoff by using line segments to cover the resources. A delay function , where is the resource type and is the number of players covering the resource, specifies the payoff for covering a resource. For gold sites, and , where so that if two players cover a resource they receive a smaller payoff. For mine sites, is a constant penalty.

#### Strategy Space:

Each player uses a function to specify which line player covers at each horizontal location. Player covers the resource at location if . The strategy space of a player with capability contains all functions with no more than segments:

 Seg(f) \raisebox−1.29pt$\tiny def=$(4M−2∑i=01f(i)≠f(i+1))+1 LC \raisebox−1.29pt$\tiny def=${f:[0,4M−1]↦{0,1}|Seg(f)≤C}

We use and to denote the capabilities of player A and B respectively, and use and for their strategies. Note that Yang et al. [5] shows that the capability bound has a natural interpretation as the size of programs in a Domain-Specific Language (DSL) describing the strategy space.

Below is our main result:

###### Theorem 3.1

Given an instance of MGMG parameterized by that satisfies , in a pure Nash equilibrium of this game, the players receive the following payoffs and :

 uA uB where C′A =min(CA,2M+1),C′B=min(CB,2M+1),t∈{0,1}

Three cases determine the value of :

• When , there are two classes of Nash equilibria distinguished by and .

• When , if and if .

• When , there is one Nash equilibrium. The above formulas give the same payoffs regardless of or .

In MGMG, when one player’s capability increases, their own payoff increases by or , but their opponent’s payoff decreases by . If both players get the same capability increment, the social welfare (i.e., the sum of their payoffs) can increase, decrease, or stay the same, depending on the sign of . If both players have the same capability , the social welfare is , which confirms Theorem 16 of Yang et al. [5] up to a constant bias because in MGMG we remove the last gold site on each line to simplify our analysis.

In the following two sections, we first present three lemmas that characterize the Nash equilibria in MGMG, and then derive the above results based on these lemmas.

### 3.1 Characteristics of Nash equilibria

We introduce some notation:

• A pair denotes a strategy profile, i.e., the strategies of both players.

• Given a strategy profile, and are the payoffs of individual players.

• Given a strategy , and denote the locations and numbers of gold and mine sites covered by the strategy:

 G(f) \raisebox−1.29pt$\tiny def=${4i∣∣0≤i
• Given a strategy for player , discontinuity points (DPs) are the locations where changes the line that covers. We also differentiate between upward discontinuity points (UDPs, denoted by ) and downward discontinuity points (DDPs, denoted by ):

 D↑(f) \raisebox−1.29pt$\tiny def=${i∣∣0≤i≤4M−2 and f(i)=0 and f(i+1)=1} D↓(f) \raisebox−1.29pt$\tiny def=${i∣∣0≤i≤4M−2 and f(i)=1 and f(i+1)=0}

Note that .

• A strategy is a perfect cover for resources located between if all gold sites are covered and all mine sites are avoided: and . We also call the resources perfectly covered in this case, and imperfectly covered otherwise. Here denotes all integers in the interval: . Note that to perfectly cover resources for , one needs segments.

• Strict strategy spaces use exactly the given number of segments:

 ˜L1 \raisebox−1.29pt$\tiny def=$L1 ˜LC \raisebox−1.29pt$\tiny def=$LC−LC−1 ={f:[0,4M−1]↦{0,1}|Seg(f)=C}
• A strategy profile is a complete-gold-coverage for a MGMG if both players cover all gold sites together, i.e., .

First, we show that DPs only occur at certain locations:

###### Lemma 1

Let be a best response of a player given the other player’s strategy. Upward discontinuity points in occur only at neighboring mine sites, and downward discontinuity points occur only at neighboring gold sites:

 ∀i∈D↑(f) :imod4=2 ∀i∈D↓(f) :imod4=0
###### Proof

We consider cases in a local region for different values of and .

• : Figure 2 shows the cases with one DP. The payoffs of covered gold sites are denoted as and , which can be or depending on the opponent’s strategy. Clearly, the payoff is maximized only when the DP is at location 2 modulo 4. It can be verified that using more DPs while maintaining and does not improve payoff.

• : Similarly, the best response in this case has one DDP at location 0.

• : The best response should have no DP. If there are DPs, there should be one UDP and one DDP to cover one gold site and no mine site, but moving the DDP rightward to also cover the gold at gives better payoff with the same number of segments.

• : The best response should either have no DP (covering two gold sites and one mine site) or two DPs (covering three gold sites and no mine site) at locations 0 and 2 modulo 4.

Now we show that the number of gold and mine sites covered by an optimal strategy is fairly predictable, i.e., it only depends on and :

###### Lemma 2

If is a strategy that conforms to Lemma 1, then

 #g(f) =M+⌊Seg(f)+f(0)−12⌋ #m(f) =M−⌊Seg(f)−f(0)2⌋
###### Proof

We define a series of strategies . Let and define the strategy obtained by removing the last DP of , i.e., where . Lemma 1 implies that each DDP adds an extra gold site and each UDP avoids a mine site, which means either or , depending on whether the last DP of is DDP or UDP. It follows that and for any pair .

We first assume . In this case, since UDPs and DDPs are interleaving, we have and . Let . We also know that since covers exactly one line. Therefore, and . A similar analysis for the case gives and . Lemma 2 summarizes these results compactly.

Now let’s shift our attention from one player’s strategy to a strategy profile.

###### Lemma 3

If and is a pure Nash equilibrium when players are limited to the strict strategy spaces and , then is a complete-gold-coverage.

###### Proof

We prove this statement in two steps. We first show that for any player, their payoff is maximized when they cover as many unoccupied gold sites as possible. Then we show that complete-gold-coverage is always feasible.

Let be the total number of gold sites covered by a strategy profile: .

Without loss of generality, we focus on player A. We show that if there is a strategy such that where , then is not a Nash equilibrium because A can get better payoff by switching to . Note that the number of gold sites covered by both players in the strategy profile is , while the number of gold sites covered by A exclusively is , which implies:

 uA(fA,fB) =(T(fA,fB)−#g(fB))⋅1 +(#g(fA)+#g(fB)−T(fA,fB))⋅ρ+#m(fA)⋅μ

Substituting the results of Lemma 2 into the above:

 uA(fA,fB) =ρ⌊CA+fA(0)−12⌋+(−μ)⌊CA−fA(0)2⌋+(1−ρ)T(fA,fB) +(ρ−1)#g(fB)+(ρ+μ)M

Let be the first two terms. One can verify that for all possible values of , , and . Note implies . Thus .

 uA(f′A,fB)−uA(fA,fB) =h(f′A)−h(fA)+(T(f′A,fB)−T(fA,fB))(1−ρ) ≥(ρ+μ)+(1−ρ) >0

Therefore, A first maximizes and then maximizes in their best response. The maximum possible value of is which is achieved when is a complete-gold-coverage.

Next we show that complete-gold-coverage is always feasible. We assume WLOG. For any strategy played by player B that conforms to Lemma 1, we show that there exists such that .

If , then A can cover all gold sites trivially. Now we consider the case . We first construct a strategy that may or may not use all the segments. For , we set and . When , we use the same value for and ; otherwise we add one discontinuity point at or according to Lemma 1. Note that Lemma 1 also implies . It is easy to verify that and . We then derive from using Algorithm 1 so that .

### 3.2 Capability Transfer Function of MGMG

Recall that in the proof of Lemma 3, we have shown that given a strategy profile that is a complete-gold-coverage, A’s payoff is

 uA(fA,fB) =ρ⌊Seg(fA)+fA(0)−12⌋+(−μ)⌊Seg(fA)−fA(0)2⌋ +(ρ−1)#g(fB)+(μ−ρ+2)M

For two strategy profiles and that are both complete-gold-coverage, we make the following two observations that can be verified using the above expansion of :

1. If and , then .

2. If , , , and , then .

In other words, the best strategy of player given the strategy of the other player satisfies:

1. is a complete-gold-coverage.

2. uses the full capability of player up to line segments, i.e., .

3. If there is a strategy that starts at line 0 (i.e., ) and satisfies both of the above constraints, then player plays such a strategy.

Next we derive the capability transfer function for the different cases. We assume WLOG:

• : All resources are perfectly covered by both players. They receive the same payoff of .

• : A perfectly covers all resources. B starts at and uses all their capability.

 #g(fB) =M+⌊CB−12⌋ uA =2M+(ρ−1)#g(fB) =(ρ+1)M+(ρ−1)⌊CB−12⌋ uB =⌊CB−12⌋ρ−⌊CB2⌋μ+(ρ+μ)M
• : Let B first play an arbitrary strategy . If , A will set to ensure a complete-gold-coverage; otherwise if , A will set due to the second observation noted above. A can derive one of their best response according to the proof of Lemma 3. Following a similar reasoning from B’s perspective, it can be shown that is also a best response of B given A’s strategy . Therefore, is a Nash equilibrium. There are two different classes of Nash equilibria: one with and , and the other with and . Let and . We have:

 uA =⌊CA+t−12⌋ρ−⌊CA−t2⌋μ+⌊CB−t2⌋(ρ−1)+(μ+1)M uB =⌊CB−t2⌋ρ−⌊CB+t−12⌋μ+⌊CA+t−12⌋(ρ−1)+(μ+1)M t ∈{0,1}

Theorem 3.1 summarizes the results of these three cases.

## References

• [1] J. F. Nash Jr (1950) Equilibrium points in n-person games. Proceedings of the national academy of sciences 36 (1), pp. 48–49. Cited by: §2.
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• [5] Y. Yang, K. Jia, and M. Rinard (2022) On the impact of player capability on congestion games. arXiv preprint arXiv:2205.09905. Cited by: Mixed Capability Games, §1, §3, §3, §3, Example 1.