A Logical–Computational Framework for Discovering Three-Player Games with Unique Pure Nash Equilibrium Payoffs

Abstract

The Nash equilibrium is a central concept in game theory, widely used across economics, social sciences, computer science, and artificial intelligence. However, computing Nash equilibria, especially in multi-player games, is a complex and computationally challenging task. Among the various types of Nash equilibria, the unique pure-strategy Nash equilibrium payoffs possess particularly desirable properties that make them suitable for deeper analysis and application. In this paper, we propose a first-order logical framework for three-player finite games, inspired by the notion of Pareto optimality, to identify a class of games with unique pure-strategy Nash equilibrium payoffs. By utilizing a SAT solver and the finite verifiability of ternary clauses, we automatically discover several families of three-player games that exhibit unique pure-strategy Nash equilibrium payoffs. This approach provides new insights into the computational aspects of game theory and offers an automated method for discovering novel game-theoretic structures.

1. Introduction

Few theories have reshaped modern economics and decision sciences as profoundly as game theory, whose influence now spans fields from political bargaining and market design to evolutionary biology and artificial intelligence [1]. As a core concept in game theory, the Nash equilibrium provides a fundamental framework for analyzing strategic interactions. Although Nash [2] proved that any finite game has a (possibly mixed-strategy) Nash equilibrium, solving or analyzing the properties of equilibria in multi-player games (such as Pareto optimality Nash equilibrium [3] and consistency with Stackelberg equilibrium [4]) is generally very difficult [5]. The uniqueness of the pure Nash equilibrium (PNE) has received special attention because of the clarity and stability of its predictions. For example, the asymmetric rent-seeking game designed by Yamazaki [6] has been shown to have a unique PNE. Meanwhile, the uniqueness of pure Nash equilibrium (PNE) payoffs is equally interesting. The concept of the “unique payoff equilibrium” proposed by Baron (allowing multiple equilibrium points with the same payoff) has been applied to network traffic allocation to avoid congestion and improve robustness [7,8].

In this context, a systematic exploration of game models possessing unique PNE payoffs becomes crucial. Early research has already made progress in this regard. Tang and Lin [9,10] discovered multiple two-player game models with unique PNE payoffs through computer-aided theorem-proving. In particular, in strictly competitive two-player games, if there are PNEs, their payoffs must be the same [11], and these equilibria are all Pareto-optimal [12]. However, many key interactions in reality involve three or more participants (such as traffic governance [13] and public resource allocation [14]). As the basis for understanding multi-player interactions, three-player games introduce core elements such as coalition formation, the possibility of collusion, and more complex externalities [15]. In-depth research on their equilibrium characteristics, especially the existence condition for unique PNE payoffs, is urgently needed.

Currently, it remains unclear what types of three-player games can guarantee that their PNEs (if they exist) have unique payoffs. Inspired by the fact that PNEs in strictly competitive two-player games must be Pareto-optimal [12], we explore whether Pareto-optimal PNEs are sufficient to ensure unique payoffs. Unfortunately, this is not a sufficient condition. To this end, we propose a stronger equilibrium concept: strictly Pareto-optimal three-player games. In such games, for any PNE, if any player unilaterally changes their strategy to increase (including keeping unchanged) their own payoffs, it will inevitably lead to a decrease (including keeping them unchanged) in the payoffs of all other players. We rigorously prove that PNEs in strictly Pareto-optimal three-player games must have unique payoffs. This is the first core contribution of this paper.

The second core contribution of this paper is the systematic discovery of such games using computer-aided methods. To explore more possible structures of three-player games that can generate unique PNE payoffs, we formalize the game as a first-order logic theory, following the approach of Tang et al. [9,10,16]. Specifically, we generate first-order logic statements similar in structure to strictly Pareto-optimal games and use a SAT solver (satisfiability problem solver) [17], a powerful automated reasoning tool, to efficiently verify whether the games defined by these statements satisfy the condition of unique PNE payoffs. The successful verification of computational experiments provides key clues for subsequent theoretical analysis.

Based on the results of computational discovery, we conduct theoretical induction and proof, identifying two new types of three-player games that guarantee unique PNE payoffs: strictly Pareto-competitive games and dominant-competitive games. Both types embody the core ideas of strictly competitive two-player games [18], which represent the third core contribution of this paper.

The structure of this paper is as follows: Section 2 introduces the basic concepts of three-player games and the formal definition of strictly Pareto-optimal games. Section 3 details the first-order logic formalization of related concepts. Section 4 rigorously proves that strictly Pareto-optimal games result in unique PNE payoffs. Section 5 presents a computational verification method using SAT solver to identify games with unique PNE payoffs, leading to the theoretical definition of two new game classes. The final section presents the conclusions.

Although this paper focuses on constructing a formal theoretical framework, the study of the uniqueness of payoffs in pure Nash equilibria holds profound practical implications across various real-world contexts. This property means that even if multiple equilibria exist in a game, all equilibria result in exactly the same payoff distribution. For instance, in economics, this provides stronger predictive power for analyzing oligopolistic competition: even if firms may adopt different strategic postures (such as varying price or output combinations), as long as these postures are equilibria, the resulting profit distribution is uniquely determined. This eliminates the payoff uncertainty arising from equilibrium selection. For example, in coordination games with symmetric payoffs [19], under specific configurations, the payoffs of their equilibria are unique. In international agreements or cooperative negotiations, the uniqueness of payoffs ensures that regardless of the specific form of stable agreement reached, the fundamental gains for all parties are clear and uncontested. This provides a theoretical foundation for constructing robust cooperative frameworks. The pursuit of this deterministic outcome aligns with the core spirit of Nash’s bargaining theory, which seeks a unique solution through axiomatization [20].

Another critical application lies in algorithm and mechanism design. In distributed computing or network resource allocation, system designers often face the “equilibrium selection” problem. If a system can be proven to possess the property of “pure Nash equilibria with unique payoffs”, the design objective can be simplified to guiding the system toward any one equilibrium, because regardless of which equilibrium is reached, the final resource allocation efficiency (or cost-sharing outcome) is unique and predictable. This significantly reduces the complexity of mechanism design, offering an ideal condition for addressing the problem of “payoff-irrelevant equilibrium selection” [21]. The characterization of this property in our study provides a refined theoretical tool for identifying and designing systems with such strong stability.

2. Three-Player Games

A three-player game (in strategic form) is a triple $\mathcal{G}=\langle N,{\left(S_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N}\rangle$, where $N=\{1,2,3\}$ is the set of players; each $S_i(1\le i\le 3)$ is the strategy set of player $1,2,3$, respectively; and every $u_i(1\le i\le 3)$ is a mapping from $S_1\times S_2\times S_3$ to $\mathbb{R}$, representing the utility function or payoff function. $S_1\times S_2\times S_3$ is the strategy space of the game and each element of $S_1\times S_2\times S_3$ is called a (strategy) profile (of $\mathcal{G}$). Each rational player in a game always tries to maximize his/her utility or payoff. For a given profile $s=(s_1,s_2,\dots ,s_n)$, by $s_{-i}$, we mean $(s_1,\dots ,s_{i-1},s_{i+1},\dots ,s_n) (1\le i\le n)$.

A profile s of $\mathcal{G}$ is Pareto-optimal if there is no profile $s^{\prime }$ of $\mathcal{G}$ such that $u_i\left(s\right)\le u_i\left(s^{\prime }\right)$ for all $i\in \{1,2,3\}$ and $u_i\left(s\right)<u_i\left(s^{\prime }\right)$ for at least one player $i\in \{1,2,3\}$. Intuitively, a profile is considered Pareto-optimal if there is no alternative profile that would make some players better off without making anyone worse off.

A Pareto-optimal s of $\mathcal{G}$ is strict if for any profile $s^{\prime }$ of $\mathcal{G}$ and any player i, the following conditions holds:

$$\begin{array}{c}\hfill u_i\left(s\right)\le u_i\left(s^{\prime }\right) \mathrm{implies}, \mathrm{for} \mathrm{any} \mathrm{other} \mathrm{player} j, u_j\left(s^{\prime }\right)\le u_j\left(s\right).\end{array}$$

(1)

Alternatively, for each player $i(i=1,2,3)$, the utility function $u_i$ can be represented by the total preference relation ${\le }_i$: for any $s, s^{\prime }\in S_1\times S_2\times S_3$,

$$s{\le }_i{s}^{\prime }\mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} u_i\left(s\right)\le u_i\left(s^{\prime }\right).$$

(2)

Please note that a binary relation R (over some set S) is a total if, for any $s, s^{\prime }\in S$, either $(s, s^{\prime })\in R$ or $(s^{\prime }, s)\in R$. It is a preference relation if it is reflexive and transitive. For convenience, a three-player game is usually written as $(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$, where $S_1,S_2$, and $S_3$ is the strategy set of each player $1,2,3$, respectively, and ${\le }_1,{\le }_2,{\le }_3$ is the preference relation of each player $1,2,3$, respectively.

Two profiles, s and $s^{\prime }$ of $\mathcal{G}$, are (payoff) equivalent if, for every $i=1,2,3$,

$$s{\le }_i{s}^{\prime } \mathrm{and} s^{\prime } {\le }_{i}s.$$

(3)

Given a three-player game $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$, $x\in S_1,y\in S_2,z\in S_3$, we define the set $B_i(.,.)$ of the best responses of each player i as follows:

$$\begin{array}{c}\hfill B_1(y,z)=\left\{x\in S_1\mid \mathrm{for} \mathrm{all} x^{\prime }\in S_1,(x^{\prime }, y, z){\le }_1(x, y, z)\right\},\end{array}$$

(4)

$$\begin{array}{c}\hfill B_2(x,z)=\left\{y\in S_2\mid \mathrm{for} \mathrm{all} y^{\prime }\in S_2, (x, y^{\prime }, z){\le }_2(x, y, z)\right\},\end{array}$$

(5)

$$\begin{array}{c}\hfill B_3(x,y)=\left\{z\in S_3\mid \mathrm{for} \mathrm{all} z^{\prime }\in S_3, (x, y, z^{\prime }){\le }_3(x, y, z)\right\}.\end{array}$$

(6)

A profile $(x,y,z)\in S_1\times S_2\times S_3$ of the game $\mathcal{G}$ is called a PNE if $x\in B_1(y,z)$, $y\in B_2(x,z)$, and $z\in B_3(x,y)$.

Intuitively, a PNE is a stable strategy profile: no player would want to change his strategy if he knew what strategies the other agents were following. In other words, no player is able to increase their payoff through unilateral action (choosing another strategy while everybody else remains the same).

3. Three-Player Games in First-Order Logic

In this section, we present a three-player game in a sorted first-order language $\mathcal{L}$ with sorts $\alpha ,\beta$, and $\gamma$, respectively. The signature of $\mathcal{L}$ contains the following:

• Predicates ${\le }_i,{<}_i,{\simeq }_i(i=1,2,3)$ of the type $(\alpha \times \beta \times \gamma )\times (\alpha \times \beta \times \gamma )$ (in first-order logic, the preference relations for participants 1, 2, and 3 are treated as predicates);
• Variables $x,x^{\prime },x_1,x_1^{\prime },x_2,x_2^{\prime },\dots ,$ of sort $\alpha$;
• Constants $a_1,a_2,\dots ,$ of sort $\alpha$;
• Variables $y,y^{\prime },y_1,y_1^{\prime },y_2,y_2^{\prime },\dots ,$ of sort $\beta$;
• Constants $b_1,b_2,\dots ,$ of sort $\beta$;
• Variables $z,z^{\prime },z_1,z_1^{\prime },z_2,z_2^{\prime },\dots ,$ of sort $\gamma$;
• Constants $c_1,c_2,\dots ,$ of sort $\gamma$;
• The usual connectives $\neg ,\wedge ,\vee ,\supset ,\leftrightarrow$ for negation, conjunction, disjunction, implication, and equivalence, respectively.

And in the rest of the paper, unless otherwise stated, all free variables in a displayed formula are assumed to be universally quantified from outside.

A three-player game $\mathcal{G}$ can be represented as the theory ${\Sigma }_{\mathcal{G}}$ consisting of, for $i=1,2,3$,

$$\begin{array}{l}(x,y,z){\le }_i(x,y,z),\hfill \end{array}$$

(7)

$$\begin{array}{l}(x_1,y_1,z_1){\le }_i(x_2,y_2,z_2)\vee (x_2,y_2,z_2){\le }_i(x_1,y_1,z_1),\hfill \end{array}$$

(8)

$$\begin{array}{ll}\hfill & (x_1,y_1,z_1){\le }_i(x_2,y_2,z_2)\wedge (x_2,y_2,z_2){\le }_i(x_3,y_3,z_3)\supset (x_1,y_1,z_1){\le }_i(x_3,y_3,z_3).\hfill \end{array}$$

(9)

It is evident that ${\Sigma }_{\mathcal{G}}$ captures that every ${\le }_i(i=1,2,3)$ is a reflexive, transitive, and total order. Thus, every (finite) model of ${\Sigma }_{\mathcal{G}}$ is indeed a (finite) three-player game. For simplicity, we denote, for every $i(i=1,2,3)$,

$$\begin{array}{c}\hfill (x_1,y_1,z_1){\le }_i(x_2,y_2,z_2)\wedge (x_2,y_2,z_2){\le }_i(x_1,y_1,z_1)\end{array}$$

(10)

by $(x_1,y_1,z_1){\simeq }_i(x_2,y_2,z_2)$, and

$$\begin{array}{c}\hfill (x_1,y_1,z_1){\le }_i(x_2,y_2,z_2)\wedge \neg \left((x_2,y_2,z_2){\le }_i(x_1,y_1,z_1)\right).\end{array}$$

(11)

by $(x_1,y_1,z_1){<}_i(x_2,y_2,z_2)$.

Simultaneously, a PNE is expressed in first-order language by the below formula. That is, a profile $(\xi ,\zeta ,\eta )\in S_1\times S_2\times S_3$ is a PNE when it captures the following formula:

$$\begin{array}{cc}\hfill & \forall x\in S_1.\left((x,\zeta ,\eta ){\le }_1(\xi ,\zeta ,\eta )\right)\wedge \forall y\in S_2.\left((\xi ,y,\eta ){\le }_2(\xi ,\zeta ,\eta )\right)\wedge \hfill \\ \hfill & \forall z\in S_3.\left((\xi ,\zeta ,z){\le }_3(\xi ,\zeta ,\eta )\right).\hfill \end{array}$$

(12)

We denote the above formula by $NE(\xi ,\zeta ,\eta )$. And the uniqueness of PNE payoffs in a three-player game is expressed by:

$$\begin{array}{cc}\hfill & NE(x_1,y_1,z_1)\wedge NE(x_2,y_2,z_2)\supset \hfill \\ & (x_1,y_1,z_1){\simeq }_1(x_2,y_2,z_2)\wedge (x_1,y_1,z_1){\simeq }_2(x_2,y_2,z_2)\wedge (x_1,y_1,z_1){\simeq }_3(x_2,y_2,z_2).\hfill \end{array}$$

(13)

In the following, we will write Formula (13) as $\lambda$. We also note the following property of PNEs, specifically their preservation in subgames. That is, if a profile is a PNE of the full game and all strategies in that profile remain available in a subgame, then it is also a PNE within that subgame.

Proposition 1.

Let $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$ be a three-player game. And ${\mathcal{G}}^{\prime }= (S_1^{\prime },S_2^{\prime },S_3^{\prime },$${\le }_1^{\prime },{\le }_2^{\prime },{\le }_3^{\prime })$ is a substructure of $\mathcal{G}$, where $S_i^{\prime }\subseteq S_i$, ${\le }_i^{\prime }$ is the restriction of ${\le }_i$ on $S_i^{\prime }$ (for $i=1,2,3$). If $(a,b,c)$ is a PNE of $\mathcal{G}$, and $(a,b,c)\in S_1^{\prime }\times S_2^{\prime }\times S_3^{\prime }$, then $(a,b,c)$ is also a PNE of ${\mathcal{G}}^{\prime }$.

Proof of Proposition 1.

$$\begin{array}{ccc}& (a,b,c)\mathrm{is}\mathrm{a}\mathrm{PNE}\mathrm{of}\mathcal{G}.\hfill & \\ & \Rightarrow \forall x\in S_1.\left((x,b,c){\le }_1(a,b,c)\right)\wedge \forall y\in S_2.\left((a,y,c){\le }_2(a,b,c)\right)\wedge \hfill & \\ & \forall z\in S_3.\left((a,b,z){\le }_3(a,b,c)\right)(\mathrm{by} \mathrm{Formula} (\mathrm{(12}\left)\right)\hfill & \\ & \Rightarrow \forall x^{\prime }\in S_1^{\prime }.\left((x^{\prime },b,c){\le }_1(a,b,c)\right)(\mathrm{by} \forall x\in S_1.\left((x,b,c){\le }_1(a,b,c)\right)\mathrm{and} S_i^{\prime }\subseteq S_i)\hfill \\ & \Rightarrow \forall x^{\prime }\in S_1^{\prime }.\left((x^{\prime },b,c){\le }_1^{\prime }(a,b,c)\right)(\mathrm{by} {\le }_i^{\prime } \mathrm{being} \mathrm{the} \mathrm{restriction} \mathrm{of} {\le }_i S_i^{\prime }\hfill & \\ & \mathrm{Similarly},\mathrm{we}\mathrm{have}\forall y^{\prime }\in S_2^{\prime }.\left((a,y^{\prime },c){\le }_2^{\prime }(a,b,c)\right)\mathrm{and}\forall z^{\prime }\in S_3^{\prime }.\left((a,b,z^{\prime }){\le }_3^{\prime }(a,b,c)\right)\hfill & \\ & \Rightarrow \forall x^{\prime }\in S_1^{\prime }.\left((x^{\prime },b,c){\le }_1^{\prime }(a,b,c)\right)\wedge \forall y^{\prime }\in S_2^{\prime }.\left((a,y^{\prime },c){\le }_2^{\prime }(a,b,c)\right)\wedge \hfill & \\ & \forall c^{*}\in S_3^{\prime }.\left((a,b,z^{\prime }){\le }_3^{\prime }(a,b,c)\right)\hfill & \\ & \Rightarrow (a,b,c)\mathrm{is}\mathrm{a}\mathrm{PNE}\mathrm{of}{\mathcal{G}}^{\prime }.\hfill \end{array}$$

      □

4. Strictly Pareto-Optimal Games

In a three-player game, a profile $(a,b,c)$ is strictly Pareto-optimal (SPO) if there is another profile $(x,y,z)$ that makes any player prefer this profile, as then all other players prefer $(x,y,z)$. In first-order logic, this is expressed as follows: for any player $i\in \{1,2,3\}$ and player $j,k\in \{1,2,3\}\setminus \left\{i\right\},j\ne k$:

$$\begin{array}{c}\hfill (a,b,c){\le }_i(x,y,z)\supset (x,y,z){\le }_j(a,b,c)\wedge (x,y,z){\le }_k(a,b,c).\end{array}$$

(14)

That is:

$$\begin{array}{cc}\hfill & (a,b,c){\le }_1(x,y,z)\supset (x,y,z){\le }_2(a,b,c)\wedge (x,y,z){\le }_3(a,b,c),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & (a,b,c){\le }_2(x,y,z)\supset (x,y,z){\le }_1(a,b,c)\wedge (x,y,z)){\le }_3(a,b,c),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & (a,b,c){\le }_3(x,y,z)\supset (x,y,z){\le }_1(a,b,c)\wedge (x,y,z){\le }_2(a,b,c).\hfill \end{array}$$

(17)

Definition 1

(strictly Pareto-optimal three-player game). Let $\mathcal{G}=(S_1,S_2,$$S_3,{\le }_1,{\le }_2,{\le }_3)$ be a three-player game. The game is strictly Pareto-optimal (SPO), if all PNEs of $\mathcal{G}$ are strictly Pareto-optimal (SPO).

Proposition 2.

Let $\mathcal{G}=\left(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3\right)$ be a strictly Pareto-optimal three-player game. If $\mathcal{G}$ has PNEs, then $\mathcal{G}$ has unique PNE payoffs.

Proof of Proposition 2.

$$\begin{array}{cc}\hfill & \mathrm{Let}\mathcal{G}\mathrm{be}\mathrm{strictly}\mathrm{Pareto}\mathrm{optimal}\mathrm{and}(a_1,b_1,c_1)\mathrm{and}(a_2,b_2,c_2)\mathrm{be}\mathrm{PNE}\mathrm{of}\mathcal{G}.\hfill \\ \hfill & \mathrm{For}\mathrm{convenience},\mathrm{we}\mathrm{denote}(a_1,b_1,c_1)\mathrm{as}{s}_1\mathrm{and}(a_2,b_2,c_2)\mathrm{as}{s}_2.\hfill \\ \hfill & \mathrm{Assume}\mathrm{there}\mathrm{are}\mathrm{not}\mathrm{unique}\mathrm{PNEs}\mathrm{payoffs}\mathrm{of}\mathcal{G}.\hfill \\ \hfill & \Rightarrow \mathrm{At}\mathrm{least}\mathrm{one}\mathrm{player}i\in \{1,2,3\}\mathrm{does}\mathrm{not}\mathrm{satisfy}\mathrm{Formula}\left(13\right).\hfill \\ \hfill & \Rightarrow s_1{\simeq }_i{s}_2\mathrm{is}\mathrm{false}(\mathrm{Since}{s}_1\mathrm{and}{s}_2\mathrm{are}\mathrm{both}\mathrm{PNEs}).\hfill \\ \hfill & \Rightarrow s_1{\le }_i{s}_2\mathrm{is}\mathrm{false}\mathrm{or}{s}_2{\le }_i{s}_1\mathrm{is}\mathrm{false}(\mathrm{By}\mathrm{Formula} \left(10\right)).\hfill \\ \hfill & \left(1\right)s_1{\le }_i{s}_2\mathrm{is}\mathrm{false}.\hfill \\ \hfill & \Rightarrow s_2{\le }_i{s}_1(\mathrm{Since}\mathrm{Formula} \left(8\right)).\hfill \\ \hfill & \Rightarrow s_1{\le }_j{s}_2(\mathrm{Since}{s}_2\mathrm{is}\mathrm{SPO}\mathrm{and}\mathrm{by}\mathrm{Formula} \left(14\right)\mathrm{we}\mathrm{have}\mathrm{that}\mathrm{for}\mathrm{any}\mathrm{player}i\in \{1,2,3\},\hfill \\ \hfill & \mathrm{player}j,k\in \{1,2,3\}\setminus \left\{i\right\},j\ne k,s_2{\le }_i{s}_1\supset s_1{\le }_j{s}_2\wedge s_1{\le }_k{s}_2).\hfill \\ \hfill & \Rightarrow s_2{\le }_k{s}_1(\mathrm{Since}{s}_1\mathrm{is}\mathrm{SPO}\mathrm{and}\mathrm{by}\mathrm{Formula} \left(14\right)\mathrm{we}\mathrm{have}\mathrm{that}\mathrm{for}\mathrm{any}\mathrm{player}j\in \{1,2,3\},\hfill \\ \hfill & \mathrm{player}i,k\in \{1,2,3\}\setminus \left\{j\right\},i\ne k,s_1{\le }_j{s}_2\supset s_2{\le }_i{s}_1\wedge s_2{\le }_k{s}_1).\hfill \\ \hfill & \Rightarrow s_1{\le }_i{s}_2(\mathrm{Since}{s}_2\mathrm{is}\mathrm{SPO}\mathrm{and}\mathrm{by}\mathrm{Formula} \left(14\right)\mathrm{we}\mathrm{have}\mathrm{that}\mathrm{for}\mathrm{any}\mathrm{player}k\in \{1,2,3\},\hfill \\ \hfill & \mathrm{player}i,j\in \{1,2,3\}\setminus \left\{k\right\},i\ne j,s_2{\le }_k{s}_1\supset s_1{\le }_i{s}_2\wedge s_1{\le }_i{s}_2),\hfill \\ \hfill & \mathrm{which}\mathrm{contradicts}\mathrm{to}\mathrm{that}{s}_1{\le }_i{s}_2\mathrm{is}\mathrm{false}.\hfill \\ \hfill & \left(2\right)s_2{\le }_i{s}_1\mathrm{is}\mathrm{false}.\hfill \\ \hfill & \mathrm{Similar}\mathrm{to}\mathrm{the}\mathrm{proof}\mathrm{in}\left(1\right).\hfill \\ \hfill & \Rightarrow \mathrm{There}\mathrm{are}\mathrm{unique}\mathrm{PNEs}\mathrm{payoffs}\mathrm{of}\mathcal{G}.\hfill \end{array}$$

      □

Now, if we denote the conjunction of (15), (16), and (17) as $\Omega$, and $\mathcal{G}$ can be replaced by theory ${\Sigma }_{\mathcal{G}}$, the following holds:

$${\Sigma }_{\mathcal{G}}\vDash \Omega \supset \lambda$$

(18)

Proposition 2 provides a foundational result, demonstrating that in the context of a three-player game possessing PNEs, imposing the requirement of SPO on every PNE results in a unique payoff emerging from equilibrium play.

Example 1.

There is currently a three-player game with player $A,B,C$.

• Player A’s strategies are denoted as follows:

  −

  $a_1$: $d_1$;

  −

  $a_2$: $d_2$;

  −

  $a_3$: $d_3$.

• Player B’s strategies are denoted as follows:

  −

  $b_1$: $d_1$;

  −

  $b_2$: $d_2$.

• Player C’s strategies are denoted as follows:

  −

  $c_1$: $d_1$;

  −

  $c_2$: $d_2$.

Note that each player’s payoff depends not only on their own choice but also on the choices of the other two players, reflecting the strategic interdependence among them. And their payoffs matrices are detailed below (

Table 1. Payoff matrices for three players of strictly Pareto-optimal game.

	C Chooses ${\mathit{c}}_1$		C Chooses ${\mathit{c}}_2$
$\mathit{A}\setminus \mathit{B}$	${\mathit{b}}_{\mathbf{1}}$	${\mathit{b}}_{\mathbf{2}}$	$\mathit{A}\setminus \mathit{B}$	${\mathit{b}}_{\mathbf{1}}$	${\mathit{b}}_{\mathbf{2}}$
$a_1$	$(2,0,1)$	$(2,1,1)$	$a_1$	$(1,1,1)$	$(1,1,1)$
$a_2$	$(1,1,1)$	$(1,1,1)$	$a_2$	$(1,1,1)$	$(1,1,1)$
$a_3$	$(1,1,1)$	$(1,1,1)$	$a_3$	$(2,1,1)$	$(1,1,1)$

).

We use the best response to find PNEs, some of which are as follows:

$$B_1(b_2,c_1)=\left\{a_1\right\},B_2(a_1,c_1)=\{b_1,b_2\},B_3(a_1,b_2)=\left\{c_1\right\}$$

$$B_1(b_1,c_2)=\left\{a_3\right\},B_2(a_3,c_2)=\left\{b_1\right\},B_3(a_3,b_1)=\left\{c_2\right\}$$

Based on the best response, it is easy to determine that $(a_1,b_2,c_1)$ and $(a_3,b_1,c_2)$ are the PNEs of the game. In this game, except for the strategy profiles $(a_1,b_2,c_1)$, $(a_3,b_1,c_2)$, and $(a_1,b_1,c_1)$, the payoffs for all other strategy profiles are $(1,1,1)$. First, we examine the relationship between any other strategy profile and the PNE $(a_1,b_2,c_1)$. We compare the preference relation with another PNE $(a_3,b_1,c_2)$. The payoffs of these two profiles are the same for all players; therefore, for $i=1,2,3$, both $(a_1,b_2,c_1){\le }_i(a_3,b_1,c_2)$ and $(a_3,b_1,c_2){\le }_i(a_1,b_2,c_1)$ hold, which necessarily satisfies Formulas (15) to (17). Furthermore, for $(a_1,b_1,c_1)$, we have $(a_1,b_2,c_1){\le }_2(a_1,b_1,c_1)$ and simultaneously have $(a_1,b_1,c_1){\le }_1(a_1,b_2,c_1)\wedge (a_1,b_1,c_1){\le }_3(a_1,b_2,c_1)$. For players 1 and 3, the payoffs of these two profiles are the same. Leaving aside the two profiles $(a_1,b_1,c_1)$ and $(a_3,b_1,c_2)$, when comparing $(a_1,b_2,c_1)$ with any other profile, only player 1 obtains a higher payoff in $(a_1,b_2,c_1)$, while the payoffs for all other players are identical between the two. In summary, the PNE $(a_1,b_2,c_1)$ is strictly Pareto-optimal. Similarly, $(a_3,b_1,c_2)$ is strictly Pareto-optimal. Therefore, this three-player game is SPO. At this point, $(a_1,b_2,c_1)$ and $(a_3,b_1,c_2)$ have the same payoff for each player, and the game has unique PNE payoffs.

It can be seen that if we can find a type of game with unique PNE payoffs, then we can design a reward mechanism: regardless of the strategy profile ultimately chosen by the players, the rewards for every player will remain strictly consistent. The following sections will further explore other types of games that possess a unique pure Nash equilibrium payoff.

5. Computer-Aided Theorem Discovery

We already know a condition that can lead to unique PNE payoffs. Next, we use computer-aided theorem discovery to explore whether other conditions of similar forms can also lead to this property. The core approach is to encode the game and payoff uniqueness as a Boolean satisfiability problem, automatically generate candidate conditions from a predefined hypothesis space, and employ a SAT solver for verification and filtering, thereby systematically discovering new valid conditions.

To computationally verify that a logical property like Formula (18) ensures unique PNE payoffs, we face a methodological challenge: the property must hold for games of any finite size. The following theorem provides the key, as it embodies a finite verifiable property. It states that verifying such an implication for all games is reduced to checking it only for games whose strategy set sizes are bounded by a small constant. This finite boundedness is what makes automated verification via SAT solver feasible, and we formally state this enabling result next.

Theorem 1.

Assume that Q is a formula without quantifiers, ${\overrightarrow{x}}_1$ and ${\overrightarrow{x}}_2$ are tuples of variables belonging to category α, ${\overrightarrow{y}}_1$ and ${\overrightarrow{y}}_2$ are tuples of variables belonging to category β, and ${\overrightarrow{z}}_1$ and ${\overrightarrow{z}}_2$ are tuples of variables belonging to category γ. We determine the following ($|{\overrightarrow{x}}_1|$ denotes the number of variables in ${\overrightarrow{x}}_1$, and similarly for $|{\overrightarrow{y}}_1|$, etc.):

$$\begin{array}{cc}\hfill & 1.{\Sigma }_{\mathcal{G}}\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \lambda \mathit{if}\mathit{for}\mathit{all}\mathit{models}\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)\mathrm{of}{\Sigma }_{\mathcal{G}},\mathit{such}\hfill \\ \hfill & \mathit{that}|S_1|\le |{\overrightarrow{x}}_1|+2,|S_2|\le |{\overrightarrow{y}}_1|+2\mathit{and}|S_3|\le |{\overrightarrow{z}}_1|+2,\mathit{we}\mathit{have}G\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_2\hfill \\ \hfill & Q\supset \lambda .\hfill \\ \hfill & 2.{\Sigma }_{\mathcal{G}}\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \neg NE(a,b,c),\mathit{where}Q\mathit{contains}\mathit{constants}a,b,\mathit{and}c\mathit{if}\mathit{for}\hfill \\ \hfill & \mathit{all}\mathit{models}\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)\mathrm{of}{\Sigma }_{\mathcal{G}},\mathit{such}\mathit{that}|S_1|\le |{\overrightarrow{x}}_1|+1,|S_2|\le |{\overrightarrow{y}}_1|+1\mathit{and}\hfill \\ \hfill & |S_3|\le |{\overrightarrow{z}}_1|+1,\mathit{we}\mathit{have}G\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \neg NE(a,b,c).\hfill \\ \hfill & \mathit{In}\mathit{these}\mathit{theorems},S_1\mathit{is}\mathit{the}\mathit{domain}\mathit{of}\mathit{category}\alpha \mathit{in}G,S_2\mathit{is}\mathit{the}\mathit{domain}\mathit{of}\mathit{category}\beta \mathit{in}G,S_3\hfill \\ \hfill & \mathit{is}\mathit{the}\mathit{domain}\mathit{of}\mathit{category}\gamma \mathit{in}G,\mathit{and}(a,b,c)\in S_1\times S_2\times S_3.\hfill \end{array}$$

Proof of Theorem 1.

$$\begin{array}{cc}\hfill & 1.\hfill \\ \hfill & (\Rightarrow ):‘‘\Rightarrow ’’\mathrm{part}\mathrm{is}\mathrm{obvious}.\hfill \\ \hfill & (\Leftarrow ):\mathrm{Assume}\mathrm{that}{\Sigma }_{\mathcal{G}}\cancel{\vDash } \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \lambda .\hfill \\ \hfill & \Rightarrow {\Sigma }_{\mathcal{G}}\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\mathrm{and}{\Sigma }_{\mathcal{G}}\cancel{\vDash }\lambda .\hfill \\ \hfill & \mathrm{Let}\mathcal{G}\mathrm{be}\mathrm{a}\mathrm{model}\mathrm{of}{\Sigma }_{\mathcal{G}}\mathrm{and}\mathcal{G}\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q.\mathrm{And}\mathcal{G}\mathrm{has}\mathrm{two}\mathrm{PNEs},(a,b,c)\hfill \\ \hfill & \mathrm{and}(a^{\prime },b^{\prime },c^{\prime }),\mathrm{with}\mathrm{different}\mathrm{payoffs},\mathrm{i}.\mathrm{e}.,\mathrm{either}(a,b,c){\cancel{\simeq }}_1(a^{\prime },b^{\prime },c^{\prime }),(a,b,c){\cancel{\simeq }}_2(a^{\prime },b^{\prime },c^{\prime }),\hfill \\ \hfill & \mathrm{or}(a,b,c){\cancel{\simeq }}_3(a^{\prime },b^{\prime },c^{\prime }).\hfill \\ \hfill & \Rightarrow \mathrm{There}\mathrm{exists}\mathrm{an}\mathrm{assignment}v\mathrm{with}D\mathrm{as}\mathrm{the}\mathrm{domain}\mathrm{of}\mathrm{discourse},v:V\to D,\mathrm{and}\hfill \\ \hfill & \exists {\overrightarrow{a}}_1\in S_1^{|{\overrightarrow{x}}_1|},\exists {\overrightarrow{b}}_1\in S_2^{|{\overrightarrow{y}}_1|},\exists {\overrightarrow{c}}_1\in S_3^{|{\overrightarrow{z}}_1|},\mathrm{such}\mathrm{that}\mathcal{G}\to {\mathcal{G}}^v:{\overrightarrow{x}}_1\to {\overrightarrow{a}}_1,{\overrightarrow{y}}_1\to {\overrightarrow{b}}_1\mathrm{and}{\overrightarrow{z}}_1\to {\overrightarrow{c}}_1.\hfill \\ \hfill & \mathrm{We}\mathrm{determine}\mathrm{that}{\mathcal{G}}^v\vDash \forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_2{Q}^{v({\overrightarrow{x}}_1/{\overrightarrow{a}}_1,{\overrightarrow{y}}_1/{\overrightarrow{b}}_1,{\overrightarrow{z}}_1/{\overrightarrow{c}}_1)}(\mathrm{by}G\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \lambda ).\hfill \\ \hfill & \mathrm{We}\mathrm{design}\mathrm{a}\mathrm{game}{\mathcal{G}}^{\prime }=(S_1^{\prime },S_2^{\prime },S_3^{\prime },{\le }_1^{\prime },{\le }_2^{\prime },{\le }_3^{\prime }):\hfill \\ \hfill & •S_1^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{a}}_1\cup \{a,a^{\prime }\}\},\mathrm{where}a,a^{\prime }\mathrm{is}\mathrm{not}\mathrm{in}{\overrightarrow{a}}_1.\hfill \\ \hfill & •S_2^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{b}}_1\cup \{b,b^{\prime }\}\},\mathrm{where}b,b^{\prime }\mathrm{is}\mathrm{not}\mathrm{in}{\overrightarrow{b}}_1.\hfill \\ \hfill & •S_3^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{c}}_1\cup \{c,c^{\prime }\}\},\mathrm{where}c,c^{\prime }\mathrm{is}\mathrm{not}\mathrm{in}{\overrightarrow{c}}_1.\hfill \\ \hfill & •{\le }_1^{\prime }=\left\{((a^{*},b^{*},c^{*}),(a^{\prime *},b^{\prime *},c^{\prime *}))\right|(a^{*},b^{*},c^{*})\in S_1^{\prime }\times S_2^{\prime }\times S_3^{\prime },(a^{\prime *},b^{\prime *},c^{\prime *})\in S_1^{\prime }\times S_2^{\prime }\times S_3^{\prime },\hfill \\ \hfill & ((a^{*},b^{*},c^{*}),(a^{\prime *},b^{\prime *},c^{\prime *}))\in {\le }_1\},\mathrm{and}\mathrm{similarly},{\le }_2^{\prime }\mathrm{and}{\le }_3^{\prime }\mathrm{follow}\mathrm{the}\mathrm{same}\mathrm{pattern}.\hfill \\ \hfill & \Rightarrow {\mathcal{G}}^{\prime }\vDash {\Sigma }_{\mathcal{G}}(\mathrm{since}{\mathcal{G}}^{\prime }\mathrm{is}\mathrm{a}\mathrm{substructure}\mathrm{of}G),\mathrm{where}\mathrm{the}\mathrm{size}\mathrm{of}{\mathcal{G}}^{\prime }\mathrm{is}\mathrm{smaller}\mathrm{or}\mathrm{equal}\mathrm{to}\hfill \\ \hfill & \left(\right|{\overrightarrow{x}}_1|+2)\times \left(\right|{\overrightarrow{y}}_1|+2)\times \left(\right|{\overrightarrow{z}}_1|+2).\hfill \\ \hfill & \Rightarrow {\mathcal{G}}^{\prime }\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q(\mathrm{since}{\mathcal{G}}^v\vDash \forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_2{Q}^{v({\overrightarrow{x}}_1/{\overrightarrow{a}}_1,{\overrightarrow{y}}_1/{\overrightarrow{b}}_1,{\overrightarrow{z}}_1/{\overrightarrow{c}}_1)}).\hfill \\ \hfill & \Rightarrow (a,b,c)\mathrm{and}(a^{\prime },b^{\prime },c^{\prime })\mathrm{are}\mathrm{both}\mathrm{PNEs}\mathrm{of}{\mathcal{G}}^{\prime }(\mathrm{by}\mathrm{Proposition}1).\hfill \\ \hfill & \Rightarrow {\mathcal{G}}^{\prime }\cancel{\vDash }\lambda .\hfill \\ \hfill & \Rightarrow {\mathcal{G}}^{\prime }\cancel{\vDash }\exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\supset \lambda (\mathrm{by}{\mathcal{G}}^{\prime }\vDash \exists {\overrightarrow{x}}_1\exists {\overrightarrow{y}}_1\exists {\overrightarrow{z}}_1\forall {\overrightarrow{x}}_2\forall {\overrightarrow{y}}_2\forall {\overrightarrow{z}}_{2}Q\mathrm{and}{\mathcal{G}}^{\prime }\cancel{\vDash }\lambda ),\mathrm{which}\hfill \\ \hfill & \mathrm{contradicts}\mathrm{the}\mathrm{premise}.\hfill \\ \hfill & 2.\hfill \\ \hfill & \mathrm{This}\mathrm{part}\mathrm{of}\mathrm{the}\mathrm{proof}\mathrm{is}\mathrm{similar}\mathrm{to}\mathrm{Part}1.\mathrm{We}\mathrm{just}\mathrm{replace}‘‘\mathcal{G}\mathrm{has}\mathrm{two}\mathrm{PNEs}(a,b,c),\hfill \\ \hfill & (a^{\prime },b^{\prime },c^{\prime })\mathrm{with}\mathrm{different}\mathrm{payoffs}"\mathrm{with}‘‘\mathcal{G}\mathrm{has}\mathrm{a}\mathrm{PNE}(a,b,c)’’,\mathrm{and}\mathrm{replace}\hfill \\ \hfill & ‘‘S_1^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in} {\overrightarrow{a}}_1\cup \{a,a^{\prime }\}\},\mathrm{where}a,a^{\prime }\mathrm{not}\mathrm{in}{\overrightarrow{a}}_1",\hfill \\ \hfill & ‘‘S_2^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{b}}_1\cup \{b,b^{\prime }\}\},\mathrm{where}b,b^{\prime }\mathrm{not}\mathrm{in}{\overrightarrow{b}}_1",\hfill \\ \hfill & ‘‘S_3^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{c}}_1\cup \{c,c^{\prime }\}\},\mathrm{where}c,c^{\prime }\mathrm{not}\mathrm{in}{\overrightarrow{c}}_1"\hfill \\ \hfill & \mathrm{with}\hfill \\ \hfill & ‘‘S_1^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{a}}_1\cup \left\{a\right\}\},\mathrm{where}a\mathrm{not}\mathrm{in}{\overrightarrow{a}}_1",\hfill \\ \hfill & ‘‘S_2^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{b}}_1\cup \left\{b\right\}\},\mathrm{where}b\mathrm{not}\mathrm{in}{\overrightarrow{b}}_1",\hfill \\ \hfill & ‘‘S_3^{\prime }=\left\{d\right|d\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{occurring}\mathrm{in}{\overrightarrow{c}}_1\cup \left\{c\right\}\},\mathrm{where}c\mathrm{not}\mathrm{in}{\overrightarrow{c}}_1",\mathrm{respectively}.\hfill \end{array}$$

      □

5.1. First-Order Logic Sentence Generation

Consider the following sentences, $(l_1\supset l_2\wedge l_3)\wedge (l_4\supset l_5\wedge l_6)\wedge (l_7\supset l_8\wedge l_9)$, based on the first-order logic form of an SPO PNE, where each $l_j$, $1\le j\le 9$ is a literal, i.e., either two atoms of form $(a,b,c){\le }_i(x_j,y_j,z_j)$, $(x_j,y_j,z_j){\le }_i(a,b,c)$ or the negation of two atoms $\neg \left((a,b,c){\le }_i(x_j,y_j,z_j)\right)$, $\neg \left((x_j,y_j,z_j){\le }_i(a,b,c)\right)$. Among them, $(a,b,c){\le }_i(x_j,y_j,z_j)$ represents that for $NE(a,b,c)$, and for any $x_j\in S_1$, $y_j\in S_2$, $z_j\in S_3$, we have players whose level of liking for $(x_j,y_j,z_j)$ is at least not less than $(a,b,c)$. Other literals with similar forms also have similar meanings, and adding “¬” indicates their negation. For these nine literals, each sentence can choose from the above four forms, so we can generate a total of $4^9=262144$ sentences.

However, now consider a situation where the first-order sentence $(l_1\supset l_2\wedge l_3)\wedge (l_4\supset l_5\wedge l_6)\wedge (l_7\supset l_8\wedge l_9)$ is equivalent to $((l_1\supset l_2)\wedge (l_1\supset l_3))\wedge ((l_4\supset l_5)\wedge (l_4\supset l_6))\wedge ((l_7\supset l_8)\wedge (l_7\supset l_9))$. A sentence

$$\neg \left((a,b,c){\le }_1(x_j,y_j,z_j)\right)\supset \neg \left((a,b,c){\le }_2(x_j,y_j,z_j)\right)$$

is logically equivalent to

$$\left((a,b,c){\le }_2(x_j,y_j,z_j)\right)\supset \left((a,b,c){\le }_1(x_j,y_j,z_j)\right).$$

In other words, any implication where both sides are negated can be converted into an implication without negation, and this non-negated version can appear in other sentences we output. Therefore, for the sentences where both sides of the implication are negated, we remove them. In addition, we also determine that the following sentence

$$(a,b,c){\le }_1(x_j,y_j,z_j)\supset \neg \left((a,b,c){\le }_2(x_j,y_j,z_j)\right)$$

is logically equivalent to

$$\neg \left((a,b,c){\le }_2(x_j,y_j,z_j)\right)\supset (a,b,c){\le }_1(x_j,y_j,z_j).$$

That is to say, negation only needs to be added to the equation before implication. Now only $l_1$, $l_4$, and $l_7$ can be added with negation. Therefore, we finally obtain $4^3\times 2^6=4096$ sentences.

Below, we will verify which generated sentence can meet the uniqueness of PNE payoffs.

5.2. SAT Solver Verification

A satisfiability solver (SAT solver) is a powerful tool for determining whether a logical formula in Conjunctive Normal Form (CNF) [22] is satisfiable. In this study, we employ a SAT solver as the core method for the formal verification of various relations, constraints, and equilibrium conditions in three-player games. Specifically, we use the Glucose solver, which is based on the Conflict-Driven Clause Learning (CDCL) algorithm [23], invoked via the Python 3.9.10 pysat library, to perform satisfiability checking on encoded CNF formulas.

To apply the SAT solver, we first translate the formal description of the game into CNF formulas. All atomic propositions (e.g., the preference atom $(a_1,b_1,c_1){\le }_1(a_2,b_2,c_2)$) are treated as Boolean variables. The background theory ${\Sigma }_{\mathcal{G}}$ (comprising the axioms of reflexivity, totality, and transitivity) is converted into a set of CNF clauses. For instance, the transitivity axiom $(x_1,y_1,z_1){\le }_i(x_2,y_2,z_2)\wedge (x_2,y_2,z_2){\le }_i(x_3,y_3,z_3)\supset (x_1,y_1,z_1){\le }_i(x_3,y_3,z_3)$ is equivalently represented by the clause $\neg p\vee \neg q\vee r$, where p, q, and r are the Boolean variables corresponding to the respective atomic propositions. We denote the Conjunctive Normal Form (CNF) of any first-order logic sentence obtained through this transformation as $\mathrm{CNF}(.)$.

Each sentence generated in Section 5.1 is denoted as ${\Omega }_i(a,b,c)(i=1,\dots ,4096)$, where $(a,b,c)$ is a pure-strategy Nash equilibrium under that sentence. One such statement ${\Omega }_i(a,b,c)$ takes the form:

$$\begin{array}{cc}\hfill & (a,b,c){\le }_1(x_1,y_1,z_1)\supset (a,b,c){\le }_2(x_2,y_2,z_2)\wedge (a,b,c){\le }_3(x_3,y_3,z_3)\wedge \hfill \\ \hfill & (a,b,c){\le }_2(x_4,y_4,z_4)\supset (a,b,c){\le }_1(x_5,y_5,z_5)\wedge (a,b,c){\le }_3(x_6,y_6,z_6)\wedge \hfill \\ \hfill & (a,b,c){\le }_3(x_7,y_7,z_7)\supset (a,b,c){\le }_1(x_8,y_8,z_8)\wedge (a,b,c){\le }_2(x_9,y_9,z_9).\hfill \end{array}$$

(19)

Furthermore, we define $SIM(x_1,y_1,z_1,x_2,y_2,z_2)$ as $(x_1,y_1,z_1){\simeq }_1(x_2,y_2,z_2)\wedge (x_1,y_1,z_1)$${\simeq }_2(x_2,y_2,z_2)\wedge (x_1,y_1,z_1){\simeq }_3(x_2,y_2,z_2)$, and its negation $\neg SIM(x_1,y_1,z_1,x_2,y_2,z_2)$ indicates that for at least one player, the payoffs of the two profiles differ.

We then transform the first-order logic statement ${\Omega }_i(a,b,c)$ (whose structure is given by Formula (19)) into an equivalent CNF. For example, a sentence of the form $l_1\supset (l_2\wedge l_3)$ is equivalent to the two clauses $(\neg l_1\vee l_2)$ and $(\neg l_1\vee l_3)$. The definition of a pure-strategy Nash equilibrium $NE(a,b,c)$ and $\neg SIM(a_1,b_1,c_1,a_2,b_2,c_2)$ are similarly converted to CNF.

Although SAT solvers based on the CDCL algorithm may, in general, face unpredictable solving times due to the combinatorial explosion in the number of variables and clauses, within our verification framework, Theorem 1 ensures that the verification process is confined to highly limited strategy spaces:

• Verifying ${\Sigma }_{\mathcal{G}}\vDash \Omega \supset \neg NE(a,b,c)$ requires instantiation only in a $1\times 1\times 1$ strategy space (i.e., each player has only one strategy).
• Verifying ${\Sigma }_{\mathcal{G}}\vDash \Omega \supset \lambda$ (where $\lambda$ is $NE(x_1,y_1,z_1)\wedge NE(x_2,y_2,z_2)\supset SIM(x_1,y_1,z_1,x_2,$$y_2,z_2)$) requires instantiation only in a $2\times 2\times 2$ strategy space (i.e., each player has two strategies).

Since the total number of profiles is strictly limited (to $1^3=1$ and $2^3=8$, respectively), the size of the generated CNF formulas remains polynomial. Consequently, the SAT solver can complete the determination in constant time. This design effectively avoids combinatorial explosion, guaranteeing the efficiency of the verification process both in theory and in practice.

Based on Theorem 1 and the implication relations of propositional logic, we design Algorithm 1 to verify whether all generated statements ${\Omega }_i(a,b,c)$ satisfy the property that “if PNEs exist, then their payoffs are equivalent”. Specifically, for each ${\Omega }_i(a,b,c)$, we construct the CNF Formula $F_1=\mathrm{CNF}({\Sigma }_{\mathcal{G}})\wedge \mathrm{CNF}\left({\Omega }_i(a,b,c)\right)\wedge \mathrm{CNF}\left(NE(a,b,c)\right)$.

We check the satisfiability of $F_1$ to verify whether ${\Sigma }_{\mathcal{G}}\wedge {\Omega }_i(a,b,c)\wedge NE(a,b,c)$ is satisfiable. If $F_1$ is satisfiable (SAT), it indicates that ${\Sigma }_{\mathcal{G}}\vDash {\Omega }_i(a,b,c)\supset \neg NE(a,b,c)$ is unsatisfiable. This means that the condition ${\Omega }_i(a,b,c)$ is compatible with $(a,b,c)$ being a PNE. According to our verification goal, we retain such ${\Omega }_i(a,b,c)$ for further testing because they are compatible with the assumption that a PNE exists. Conversely, if $F_1$ is unsatisfiable (UNSAT), it indicates that ${\Omega }_i(a,b,c)$ implies that $(a,b,c)$ is not a PNE, and we exclude ${\Omega }_i(a,b,c)$, as they may trivially lead to the non-existence of a PNE.

For those ${\Omega }_i(a,b,c)$ that pass this test (i.e., $F_1$ is satisfiable), we further construct the CNF Formula $F_2=\mathrm{CNF}({\Sigma }_{\mathcal{G}})\wedge \mathrm{CNF}\left({\Omega }_i(a_1,b_1,c_1)\right)\wedge \mathrm{CNF}\left({\Omega }_i(a_2,b_2,c_2)\right)\wedge \mathrm{CNF}\left(NE(a_1,b_1,c_1)\right)\wedge \mathrm{CNF}\left(NE(a_2,b_2,c_2)\right)\wedge \mathrm{CNF}(\neg SIM(a_1,b_1,c_1,a_2,b_2,c_2))$. In $F_2$, in addition to ${\Sigma }_{\mathcal{G}}$, we also include ${\Omega }_i(a_1,b_1,c_1)$ and ${\Omega }_i(a_2,b_2,c_2)$ (obtained by replacing $(a_i,b_i,c_i)$ with $(a,b,c)$ in ${\Omega }_i(a,b,c)$, for $i=1,2$), as well as the CNF representations of $NE(a_1,b_1,c_1)$, $NE(a_2,b_2,c_2)$, and $\neg SIM(a_1,b_1,c_1,a_2,b_2,c_2)$. By checking the satisfiability of $F_2$, we verify whether ${\Sigma }_{\mathcal{G}}\vDash {\Omega }_i(a,b,c)\supset \lambda$ holds. If $F_2$ is unsatisfiable (UNSAT), it indicates that ${\Sigma }_{\mathcal{G}}\vDash {\Omega }_i(a,b,c)\supset \lambda$ holds, and therefore ${\Omega }_i(a,b,c)$ guarantees that any two PNEs have the same payoff.

Algorithm 1 Unique PNE Payoff Condition ${\Omega }_i(a,b,c)$

Require:  •   ${\Omega }_i(a,b,c)$ ($i=1,\dots ,4096$, in the form of Formula (19))      •    $F_1=\mathrm{CNF}\left({\Sigma }_{\mathcal{G}}\right)\wedge \mathrm{CNF}\left({\Omega }_i(a,b,c)\right)\wedge \mathrm{CNF}\left(NE(a,b,c)\right)$      •    Two PNE profiles $(a_1,b_1,c_2)$ and $(a_2,b_2,c_2)$ Ensure:  All valid ${\Omega }_i(a,b,c)$ conditions set $\mathcal{V}$   1: $\mathcal{V}\leftarrow \varnothing$   2: for each $i=1$ to 4096 do   3:        if SAT solver determines that $F_1$ is satisfiable (SAT) then   4:            $F_2\leftarrow \mathrm{CNF}\left({\Sigma }_{\mathcal{G}}\right)\wedge \mathrm{CNF}\left({\Omega }_i(a_1,b_1,c_1)\right)\wedge \mathrm{CNF}\left({\Omega }_i(a_2,b_2,c_2)\right)\wedge \mathrm{CNF}\left(NE\right(a_1,b_1,$$c_1\left)\right)\wedge \mathrm{CNF}\left(NE(a_2,b_2,c_2)\right)\wedge \mathrm{CNF}(\neg SIM(a_1,b_1,c_1,a_2,b_2,c_2))$   5:            if SAT solver determines that $F_2$ is unsatisfiable (UNSAT) then   6:                $\mathcal{V}\leftarrow \mathcal{V}\cup \left\{{\Omega }_i(a,b,c)\right\}$   7:            end if   8:        end if   9: end for 10: return  $\mathcal{V}$

The complete implementation code and a reproduction package are publicly available (https://github.com/Nicole-Yangjia/three-player-game-with-unique-PNE-payoff (accessed on 2 December 2025)).

5.3. Game with the Unique PNE Payoffs

In the end, we obtained 2048 sentences that met unique PNE payoffs. Interestingly, we also identified conditions that contrasted with those of SPO three-player games. To encompass this alternative direction, we extend the existing definition by introducing the concept of “strictly Pareto-competitive games” to characterize these opposing conditions.

Definition 2

(strictly Pareto-competitive game). Let $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$ be a three-player game. A profile $s\in S_1\times S_2\times S_3$ is strictly Pareto-competitive if, for any other profile $s^{\prime }$ of $\mathcal{G}$, there is any player $i\in \{1,2,3\}$ such that one of the following conditions holds:

$$\begin{array}{cc}\hfill & s{\le }_i{s}^{\prime } implies s^{\prime }{\le }_{imod3+1}s and s^{\prime }{\le }_{imod3+2}s,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & s^{\prime } {\le }_{i}s implies s{\le }_{imod3+1}{s}^{\prime } and s^{\prime } {\le }_{imod3+2}{s}^{\prime }.\hfill \end{array}$$

(21)

The game $\mathcal{G}$ is strictly Pareto-competitive if all PNEs of $\mathcal{G}$ are strictly Pareto-competitive.

Proposition 3.

Let $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$ be a strictly Pareto-competitive three-player game. If $\mathcal{G}$ has PNEs, then it has unique PNE payoffs.

Proof of Proposition 3.

This part of the proof is similar to Proposition 2.

□

Since the concept of a strictly Pareto-competitive game already encompasses that of a strictly Pareto-optimal game, the game presented in Example 1 earlier is both strictly Pareto-optimal and naturally falls under the category of a strictly Pareto-competitive game. Given this conceptual inclusion, we do not provide a separate example specifically for strictly Pareto-competitive games.

In addition, we also discover other types of three-player games that can lead to unique PNE payoffs, one of which we name a dominant-competitive game.

Definition 3

(dominant-competitive three-player game). Let $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$ be a three-player game, where $s\in S_1\times S_2\times S_3$ is a PNE and $s^{\prime }\in S_1\times S_2\times S_3$ is a profile. For its three players i, j, and k, the following are true:

• Player i is positive dominant if

  $$s{\le }_j{s}^{\prime }\mathit{and}s{\le }_k{s}^{\prime }\mathit{when}s{\le }_i{s}^{\prime };$$

  (22)
• Player i is negative dominant if

  $$s^{\prime }{\le }_{j}s\mathit{and}{s}^{\prime }{\le }_{k}s\mathit{when}{s}^{\prime }{\le }_{i}s.$$

  (23)

If there exists a positive or negative dominant player i, then this game is a dominant three-player game. And this game is competitive if the following are true:

• When i is a positive dominant player and $s^{\prime }{\le }_{j}s$, we have

  $$s{\le }_k{s}^{\prime }\mathit{and}{s}^{\prime }{\le }_{i}s;$$

  (24)

  or
• When i is a negative dominant player and $s{\le }_j{s}^{\prime }$, we have

  $$s^{\prime }{\le }_{k}s\mathit{and}s{\le }_i{s}^{\prime }.$$

  (25)

Proposition 4.

Let $\mathcal{G}=(S_1,S_2,S_3,{\le }_1,{\le }_2,{\le }_3)$ be a dominant-competitive three-player game. If $\mathcal{G}$ has PNEs, then it has unique PNE payoffs.

Proof of Proposition 4.

$$\begin{array}{ccc}& \mathrm{Let}{s}_1\mathrm{and}{s}_2\mathrm{be}\mathrm{PNEs}\mathrm{of}\mathcal{G},\mathrm{player}1\mathrm{be}\mathrm{positive}\mathrm{dominant}\mathrm{player},\mathrm{and}\mathcal{G}\mathrm{be}\mathrm{a}\mathrm{dominant}\hfill & \\ & \mathrm{competitive}\mathrm{game}.\hfill & \\ & \Rightarrow s_1{\le }_1{s}_2 \mathrm{or} s_2{\le }_1{s}_1(\mathrm{by}\mathrm{Formula}\left(8\right)).\hfill & \\ & \mathrm{If}{s}_1{\le }_1{s}_2,\hfill & \\ & s_1{\le }_1{s}_2 \mathrm{implies} s_1{\le }_2{s}_2 \mathrm{and} s_1{\le }_3{s}_2(\mathrm{since}{s}_1\mathrm{is}\mathrm{a}\mathrm{PNE},\mathrm{player}1\mathrm{is}\mathrm{positive}\mathrm{dominant}\hfill & \\ & \mathrm{and}\mathrm{Formula}\left(22\right)\mathrm{also}\mathrm{confirms}\mathrm{this}).\hfill & \\ & \Rightarrow s_1{\le }_2{s}_2\hfill & \\ & s_1{\le }_2{s}_2 \mathrm{implies} s_1{\le }_1{s}_2 \mathrm{and} s_2{\le }_3{s}_1(\mathrm{aince}{s}_2\mathrm{is}\mathrm{a}\mathrm{PNE},\mathrm{player}1\mathrm{is}\mathrm{positive}\mathrm{dominant}\hfill & \\ & \mathrm{and}\mathrm{Formula}\left(24\right)\mathrm{also}\mathrm{confirms}\mathrm{this}).\hfill & \\ & \Rightarrow s_2{\le }_3{s}_1\hfill & \\ & s_2{\le }_3{s}_1 \mathrm{implies} s_2{\le }_1{s}_1 \mathrm{and} s_1{\le }_2{s}_2(\mathrm{since}{s}_2\mathrm{is}\mathrm{a}\mathrm{PNE},\mathrm{player}1\mathrm{is}\mathrm{positive}\mathrm{dominant}\hfill & \\ & \mathrm{and}\mathrm{Formula}\left(25\right)\mathrm{also}\mathrm{confirms}\mathrm{this}).\hfill & \\ & \Rightarrow s_2{\le }_1{s}_1.\hfill & \\ & \mathrm{Similarly},\mathrm{if}{s}_2{\le }_1{s}_1,\mathrm{then}{s}_1{\le }_1{s}_2.\hfill & \\ & \Rightarrow s_1{\le }_1{s}_2 \mathrm{and} s_2{\le }_1{s}_1.\hfill & \\ & \mathrm{Similarly},\mathrm{we}\mathrm{determine}\mathrm{that}{s}_1{\le }_3{s}_2,s_2{\le }_2{s}_1,s_1{\le }_3{s}_2,\mathrm{and}{s}_2{\le }_3{s}_1.\hfill & \\ & \Rightarrow \mathrm{The}\mathrm{game}\mathcal{G}\mathrm{has}\mathrm{unique}\mathrm{PNE}\mathrm{payoffs}.\hfill \end{array}$$

     □

Example 2.

There is currently a three-player game with players $A,B,C$.

• Player A’s strategies are denoted as follows:

  −

  $a_1$: $d_1$;

  −

  $a_2$: $d_2$.

• Player B’s strategies are denoted as follows:

  −

  $b_1$: $d_1$;

  −

  $b_2$: $d_2$.

• Player C’s strategies are denoted as follows:

  −

  $c_1$: $d_1$;

  −

  $c_2$: $d_2$.

This condition is similar to that described in Example 1, where each player’s payoff depends on the strategic choices of all three participants. And their payoff matrices are shown below (

Table 2. Payoff matrix for three players of dominant-competitive game.

	C Chooses ${\mathit{c}}_1$		C Chooses ${\mathit{c}}_2$
$\mathit{A}\setminus \mathit{B}$	${\mathit{b}}_{\mathbf{1}}$	${\mathit{b}}_{\mathbf{2}}$	$\mathit{A}\setminus \mathit{B}$	${\mathit{b}}_{\mathbf{1}}$	${\mathit{b}}_{\mathbf{2}}$
$a_1$	$(3,2,1)$	$(3,1,1)$	$a_1$	$(3,1,1)$	$(3,1,1)$
$a_2$	$(3,1,1)$	$(3,0,1)$	$a_2$	$(3,0,1)$	$(3,2,2)$

).

Example 2 presents a three-player dominant-competitive game in which player 1 is negative dominant. This game possesses PNEs with identical payoffs. Specifically, from the best-response correspondences:

$$B_1(b_1,c_1)=\{a_1,a_2\},B_2(a_1,c_1)=\left\{b_1\right\},B_3(a_1,b_1)=\{c_1,c_2\};$$

$$B_1(b_2,c_2)=\left\{a_2\right\},B_2(a_2,c_2)=\left\{b_2\right\},B_3(a_2,b_2)=\left\{c_2\right\},$$

we obtain two PNEs $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$. For all other strategy profiles, the payoffs are limited to only two types $(3,1,1)$ and $(3,0,1)$. We denote the profiles corresponding to these payoffs as $s_1$ and $s_2$. We consider the profile $s_1$ with payoff $(3,1,1)$. For any PNE profile s, the relations $s_1{\le }_{1}s$, $s_1{\le }_{2}s$, and $s_1{\le }_{3}s$ hold. Consequently, the condition of Formula (23) is satisfied

$$s_1{\le }_{1}s\supset s^{\prime }{\le }_{2}s\wedge s^{\prime }{\le }_{3}s.$$

Furthermore, we also have $s{\le }_3{s}_1$ and $s{\le }_1{s}_1$, which fulfill Formula (25)

$$s{\le }_3{s}_1\supset s_1{\le }_{2}s\wedge s{\le }_1{s}_1.$$

Similar conclusions can be drawn for the other non-equilibrium profile $s_2$. This confirms that Example 2 indeed constitutes a dominant-competitive three-player game with player 1 being negative dominant, and it satisfies the property of unique pure-strategy Nash equilibrium payoffs.

However, we have imposed many restrictions on the part of combination generation, such as that the form must be $(l_1\supset l_2\wedge l_3)\wedge (l_4\supset l_5\wedge l_6)\wedge (l_7\supset l_8\wedge l_9)$, and the predicate corresponding to each $l_i$ must be fixed; for example, the predicate in $l_1$ must be ${\le }_1$, which greatly reduces the scope of our verification, but may also lose some conditions that can lead to unique PNE payoffs. Therefore, if interested, you may try modifying some settings when generating sentences, such as changing all settings to $(l_1\supset l_2\vee l_3)\vee (l_4\supset l_5\vee l_6)\vee (l_7\supset l_8\vee l_9)$, and verifying the correctness of the unique PNE payoff condition according to the ideas in this article.

6. Conclusions and Future Work

This study establishes a strictly formal first-order logical framework for three-player games, enabling the systematic analysis and characterization of PNE properties. Within this framework, we for the first time systematically apply computer-aided theorem discovering to three-player games. This approach not only successfully verifies the applicability of finite verifiability properties in such games but also automatically discovers and verifies 2048 sufficient conditions that guarantee unique pure-strategy Nash equilibrium (PNE) payoffs. Through theoretical analysis of these conditions, we further distill and formally define two novel classes of games with clear intuitive interpretations, namely strictly Pareto-competitive games and dominant-competitive games. The former captures direct competitive relations among three players, while the latter focuses on asymmetric competition between two groups of players. Both classes extend beyond the classical framework of two-player strictly competitive games. The formal framework and computational methodology developed in this work provide an extensible and automatically verifiable research paradigm for systematically exploring a wider range of properties in game theory, such as equilibrium existence, stability, and efficiency. It also lays a methodological foundation for extending theoretical insights from two-player and three-player games to games with an arbitrary finite number of players.

Future research can proceed in several promising directions. The foremost task is the complete characterization of all conditions leading to unique PNE payoffs in three-player games based on the current framework, aiming to build a comprehensive classification theory. Secondly, the current model and findings can be generalized to games with any finite number of players to establish a more universal theory of multi-agent interactions. Furthermore, the proposed computational discovery framework itself can be applied to explore other equilibrium properties and solution concepts, offering new tools and perspectives for the axiomatic study of game theory.