Rationalizability is a solution concept in game theory. It is the most permissive possible solution concept that still requires both players to be at least somewhat rational and know the other players are also somewhat rational, i.e. that they do not play dominated strategies. A strategy is rationalizable if there exists some possible set of beliefs both players could have about each other's actions, that would still result in the strategy being played. Rationalizability is a broader concept than a Nash equilibrium. Both require players to respond optimally to some belief about their opponents' actions, but Nash equilibrium requires these beliefs to be correct, while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).
Definition Starting with a normal-form game, the rationalizable set of actions can be computed as follows:
Start with the full action set for each player. Remove all dominated strategies, i.e. strategies that "never make sense" (are never a best reply to any belief about the opponents' actions). The motivation for this step is no rational player would ever choose such actions. Remove all actions which are never a best reply to any belief about the opponents' remaining actions—this second step is justified because each player knows that the other players are rational. Continue the process until no further actions can be eliminated. In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
Iterated elimination of strictly dominated strategies (IESDS) The iterated elimination (or deletion, or removal) of dominated strategies (also denominated as IESDS, or IDSDS, or IRSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on. The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is common knowledge, that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976). There are two versions of this process. One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium. Moreover, iterated elimination of strictly dominated strategies is path independent. That is, if at any point in the process there are multiple strictly dominated strategies, then it doesn't matter for the end result which strategies we remove first. Strict Dominance Deletion Step-by-Step Example:
C is strictly dominated by A for Player 1. Therefore, Player 1 will never play strategy C. Player 2 knows this. (see IESDS Figure 1) Of the remaining strategies (see IESDS Figure 2), Z is strictly dominated by Y and X for Player 2. Therefore, Player 2 will never play strategy Z. Player 1 knows this. Of the remaining strategies (see IESDS Figure 3), B is strictly dominated by A for Player 1. Therefore, Player 1 will never play B. Player 2 knows this. Of the remaining strategies (see IESDS Figure 4), Y is strictly dominated by X for Player 2. Therefore, Player 2 will never play Y. Player 1 knows this. Only one rationalizable strategy is left {A,X} which results in a payoff of (10,4). This is the single Nash Equilibrium for this game. Another version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.) Weak Dominance Deletion Step-by-Step Example:
O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 5) U is weakly dominated by T for Player 2. If Player 2 chooses T, then the final equilibrium is (N,T)
O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 6) T is weakly dominated by U for Player 2. If Player 2 chooses U, then the final equilibrium is (N,U) In any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game.
Iterated elimination by mixed strategy There are instances when there is no pure strategy that dominates another pure strategy, but a mixture of two or more pure strategies can dominate another strategy. This is called Strictly Dominant Mixed Strategies. Some authors allow for elimination of strategies dominated by a mixed strategy in this way.
Example 1: In this scenario, for player 1, there is no pure strategy that dominates another pure strategy. Let's define the probability of player 1 playing up as p, and let p = 1/2. We can set a mixed strategy where player 1 plays up and down with probabilities (1/2,1/2). When player 2 plays left, then the payoff for player 1 playing the mixed strategy of up and down is 1, when player 2 plays right, the payoff for player 1 playing the mixed strategy is 0.5. Thus regardless of whether player 2 chooses left or right, player 1 gets more from playing this mixed strategy between up and down than if the player were to play the middle strategy. In this case, we should eliminate the middle strategy for player 1 since it's been dominated by the mixed strategy of playing up and down with probability (1/2,1/2).
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