Symmetric game
In game theory, a symmetric game is a strategic interaction where all players possess identical strategy sets, and the payoff structure ensures that the game appears the same from each player's perspective; specifically, if any two players exchange their chosen strategies, their payoffs are swapped while all other players' payoffs remain unchanged.[1] This strong form of symmetry, distinct from weaker variants where only non-swapping players' payoffs are unaffected, captures scenarios where players are indistinguishable in terms of available actions and incentives.[1] Symmetric games are foundational in both pure and applied game theory due to their prevalence in modeling real-world situations involving identical agents, such as competitive auctions, biological evolution, or social dilemmas. They simplify analysis by guaranteeing the existence of symmetric Nash equilibria—outcomes where all players select the same strategy—although asymmetric equilibria may also occur, as demonstrated in certain two-player examples lacking pure symmetric solutions.[2] In zero-sum symmetric games, the payoff matrix is skew-symmetric (i.e., equal to the negative transpose of itself), ensuring fairness since the value of the game is zero, with no player advantage.[3] Key examples include the Prisoner's Dilemma, where mutual defection is the unique symmetric equilibrium despite cooperative incentives, and Matching Pennies, a zero-sum game illustrating mixed-strategy symmetry.[4] Symmetric games have been central to evolutionary game theory, where they model population dynamics and strategy stability, and to multi-agent systems, enabling efficient computation of equilibria in large-scale interactions.[5] Their properties, such as ordinal symmetry (where preference orderings are preserved under transposition), further aid in classifying games and predicting behavior in symmetric environments.[4]Fundamentals
Definition
In game theory, a normal-form game (also known as a strategic-form game) consists of a finite set of players N = \{1, 2, \dots, n\}, each with a strategy set S_i representing the pure strategies available to player i, and payoff functions u_i: S \to \mathbb{R} for each player i, where S = \prod_{i \in N} S_i is the set of strategy profiles s = (s_1, \dots, s_n) with s_i \in S_i.[6][7] A symmetric game is a normal-form game in which all players have identical strategy sets, so S_i = S for all i \in N and some common S, and the payoff functions satisfy permutation invariance: for any permutation \pi: N \to N and any strategy profile s = (s_1, \dots, s_n) \in S^n, it holds that u_i(s) = u_{\pi(i)}(s_\pi) for all i \in N, where s_\pi = (s_{\pi(1)}, \dots, s_{\pi(n)}).[7] This condition ensures that the game structure remains unchanged under any relabeling of players, reflecting identical roles and incentives across participants.[7] In contrast, an asymmetric game lacks this invariance, permitting player-specific strategy sets (S_i \neq S_j for some i, j) or payoff functions that do not satisfy the permutation condition, thereby introducing distinctions in roles or outcomes based on player identity.[6][7] For the common case of two-player games, symmetry requires S_1 = S_2 = S and u_1(a, b) = u_2(b, a) for all strategies a, b \in S, meaning the payoff matrix is symmetric in the sense that row player payoffs transpose to column player payoffs.[6]Basic Examples
One of the most classic examples of a symmetric game is the Prisoner's Dilemma, a two-player scenario where each player chooses between cooperating or defecting, and the payoffs are structured such that mutual cooperation yields moderate benefits for both, but defection dominates for each individual regardless of the other's choice.[8] In this game, symmetry arises because the players are indistinguishable: swapping their roles does not alter the payoff structure, as both face identical strategy sets and mirrored outcomes based on the combination of actions chosen.[9] Another illustrative symmetric game is Rock-Paper-Scissors, a zero-sum contest where players simultaneously select one of three options—rock, paper, or scissors—each of which beats one opponent choice and loses to another in a cyclic manner.[10] The symmetry here stems from the identical strategy availability to both players and the fact that payoffs depend solely on the matchup of strategies, remaining unchanged if players are interchanged, ensuring no player-specific advantages.[11] Coordination games like the Stag Hunt also exemplify symmetry, where two players decide whether to hunt a stag (requiring mutual cooperation for a high reward) or a hare (a safer but lower-yield solo option).[12] Symmetry is evident as both players have the same strategies and receive identical payoffs for any given pair of choices, making them fully interchangeable without affecting the game's structure.[13] In contrast, the Battle of the Sexes serves as a non-example of a symmetric game, depicting a couple coordinating on an evening activity where one prefers the opera and the other the football game, leading to player-specific payoff preferences that break interchangeability.[14] This asymmetry highlights how symmetric games require that identical strategies by interchangeable players yield mirrored payoffs, a condition not met here due to differing individual incentives.[15]Symmetry in Payoff Structures
Symmetry in 2x2 Games
In symmetric 2x2 games, the payoff structure is represented using a bimatrix where both players share the same strategy set, typically labeled as actions 1 and 2. The payoff for player 1 when choosing strategy i against player 2's strategy j is denoted u_1(i,j), and the bimatrix takes the form where player 1's payoff matrix is \begin{pmatrix} a & b \\ c & d \end{pmatrix} and player 2's is \begin{pmatrix} a & c \\ b & d \end{pmatrix}, ensuring the symmetry condition holds across players.[16] This representation reduces the number of independent parameters to four (a, b, c, d), as opposed to eight in a general asymmetric 2x2 bimatrix, highlighting how symmetry imposes structural constraints on outcomes.[16] Algebraically, symmetry in these games requires that the payoff functions satisfy u_1(i,j) = u_2(j,i) for all strategy pairs (i,j), meaning the payoff player 1 receives from (i,j) equals what player 2 would receive if roles were swapped to (j,i). Visually, this manifests in the bimatrix as the off-diagonal elements mirroring across the main diagonal of player 1's matrix (i.e., the b and c entries are swapped in player 2's matrix), while diagonal elements remain identical. A special case of "pure symmetry" occurs when b = c, yielding a form like \begin{pmatrix} a & b \\ b & d \end{pmatrix} for player 1 (and its transpose for player 2), which often aligns with identical payoffs for both players.[17][16] Symmetric 2x2 games can be classified into types based on the balance between shared and opposing interests, often via decomposition of the payoff matrix into a cooperative (common interest) component and a zero-sum (conflicting interest) component. Common interest games emphasize alignment, where players prefer the same outcomes, as in coordination scenarios with payoff matrices like \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} for player 1 (transpose for player 2), both favoring diagonal matches. Conflicting interest games feature opposition, resembling zero-sum structures such as the skew-symmetric form \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} for player 1 (negative for player 2), where gains for one directly offset the other, though 2x2 cases are limited to trivial equilibria at zero. Mixed types, like the Chicken game, combine elements with tension between cooperation and defection, exemplified by player 1's matrix \begin{pmatrix} 0 & 3 \\ 2 & 1 \end{pmatrix} (transpose for player 2), where swerving avoids mutual loss but yielding concedes advantage.[18][18][19] The study of symmetry in 2x2 games traces back to early game theory, particularly von Neumann's 1928 minimax theorem, which established optimal strategies in symmetric zero-sum contexts by proving the equality of maximin and minimax values, laying foundational insights for broader symmetric analyses in the 1930s.[20]General Symmetric Payoff Matrices
In general finite symmetric games, two players each have an identical finite set of strategies, denoted as S = \{1, 2, \dots, n\}, and the payoff to player 1 choosing strategy i when player 2 chooses j is given by the entry A_{ij} in a payoff matrix A. The symmetry condition requires that u_1(i,j) = u_2(j,i) for all i, j \in S, so player 2's payoff matrix is A^T. A special case occurs when A = A^T, in which both players have the same symmetric payoff matrix, ensuring payoffs depend only on the strategies selected regardless of who selects them. In zero-sum symmetric games, an additional condition holds: u_1(i,j) + u_2(i,j) = 0, leading to A = -A^T, so player 2's payoffs are the negative of player 1's.[21] A representative example is the generalized Rock-Paper-Scissors game, a symmetric 3x3 zero-sum game where strategies correspond to Rock (1), Paper (2), and Scissors (3), with cyclic dominance. The payoff matrix A for player 1 (with player 2 receiving payoffs from -A) can be structured as follows, where wins yield +1, losses -1, and ties 0:| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0 | -1 | 1 |
| Paper | 1 | 0 | -1 |
| Scissors | -1 | 1 | 0 |