“The Nash Equilibrium” is the most common way to define the solution of a non-cooperative game involving two or more players, such as “The Prisoner’s Dilemma”. In non-cooperative games, “Nash Equilibrium” occurs when within a game a set of players have a chosen strategy that they would not want to deviate from based on the strategies of the other participating players.
Turocy and von Stengel provide an illustration of “The Prisoner’s Dilemma” in Game Theory: “The Prisoner’s Dilemma is a game in strategic form between two players. Each player has two strategies, called “cooperate” and “defect,” which are labeled C and D for player I and c and d for player II, respectively. (For simpler identification, upper case letters are used for strategies of player I and lower case letters for player II.) Figure 1 shows the resulting payoffs in this game. Player I chooses a row, either C or D, and simultaneously player II chooses one of the columns c or d. The strategy combination (C, c) has payoff 2 for each player, and the combination (D, d) gives each player payoff 1. The combination (C, d) results in payoff 0 for player I and 3 for player II, and when (D, c) is played, player I gets 3 and player II gets 0.”
“Any two-player game in strategic form can be described by a table like the one in Figure 1, with rows representing the strategies of player I and columns those of player II. (A player may have more than two strategies.) Each strategy combination defines a payoff pair, like (3, 0) for (D, c), which is given in the respective table entry. Each cell of the table shows the payoff to player I at the (lower) left, and the payoff to player II at the (right) top. These staggered payoffs, due to Thomas Schelling, also make transparent when, as here, the game is symmetric between the two players. Symmetry means that the game stays the same when the players are exchanged, corresponding to a reflection along the diagonal shown as a dotted line in Figure 2. Note that in the strategic form, there is no order between player I and II since they act simultaneously (that is, without knowing the other’s action), which makes the symmetry possible.” – Turocy and von Stengel
Turocy and von Stengel continue: “Figure 2 shows the game of Figure 1 with annotations, implied by the payoff structure. The dotted line shows the symmetry of the game. The arrows at the left and right point to the preferred strategy of player I when player II plays the left or right column, respectively. Similarly, the arrows at the top and bottom point to the preferred strategy of player II when player I plays top or bottom. In the Prisoner’s Dilemma game, “defect” is a strategy that dominates “cooperate.” Strategy D of player I dominates C since if player II chooses c, then player I’s payoff is 3 when choosing D and 2 when choosing C; if player II chooses d, then player I receives 1 for D as opposed to 0 for C. These preferences of player I are indicated by the downward-pointing arrows in Figure 2. Hence, D is indeed always better and dominates C. In the same way, strategy d dominates c for player II. No rational player will choose a dominated strategy since the player will always be better off when changing to the strategy that dominates it. The unique outcome in this game, as recommended to utility-maximizing players, is therefore (D, d) with payoffs (1, 1). Somewhat paradoxically, this is less than the payoff (2, 2) that would be achieved when the players chose (C, c).“
But, Where Is “The Nash Equilibrium” In This Game?
Let’s remember that non-cooperative games, such as “The Prisoner’s Dilemma,” have a solution if there exists an equilibrium pair among the jointly admissible strategy pairs and all jointly admissible equilibrium pairs are both interchangeable and equivalent. An equilibrium pair is a pair of strategies for the two players such that each is the player’s best strategy (or as good as any other) that can be coupled with the other’s. Equilibrium pairs are equivalent if, for each player separately, they yield equal payoffs; Equilibrium pairs are interchangeable if all pairs formed from the corresponding strategies are also equilibrium points.
In the case of “The Prisoner’s Dilemma”, the Nash equilibrium lies where players both betray each other, in the players protecting oneself from being punished more. “This solution, of course, is inefficient for the two players.” – Thomas C. Schelling. Therefore, “The Nash Equilibrium” of “The Prisoner’s Dilemma” is (D, d) with payoffs (1, 1). Interestingly, this is less than the payoff (2, 2) achieved when the players choose (C, c), had they been able to communicate for cooperation. “The Nash Equilibrium” of “The Prisoner’s Dilemma” is shown below in Figure 3:
Final Thoughts
According to Nash, “Every finite game has an equilibrium point,” to which Holt and Roth add, “What the Nash equilibrium makes clear is that the cooperative outcome, because it is not an equilibrium, is going to be unstable in ways that can make cooperation difficult to maintain.”
This combination of quotes seems to vaguely define human behavior – Life is finite, so must, by definition, have a Nash Equilibrium. But, while that may be true, at the same time, the perfect outcome to Life requires cooperation – which is difficult to maintain because it is not the equilibrium solution. In other words, people tend towards selfishness and savagery, rather than altruism and cooperation.
By working together, in the spirit of harmony, we can achieve more than if we work alone; by working together we can achieve win-win results, rather than results at the cost of someone else’s opportunity.
Thanks for reading!