Lecture 2: Main Definitions and Assumptions. Best Response, Nash Equilibrium, and Dominant Strategy Equilibrium. Iterated Elimination of Dominated Strategies Vera Sharunova 1 Prisoners’ Dilemma Suppose that there are two suspects in an armed robbery case. The suspects got caught in the get-away vehicle, but managed to toss their guns into a storm drain before being apprehended. The detectives place the two suspects in separate interrogation rooms and outline the “deal” for each suspect: 1. If neither of the suspects confesses to having a gun during the robbery, they are both charged with just the robbery and are looking at jail time of 2 years. 2. If one of the suspects confesses to having a gun, the police will set the confessor free and the other will serve substantial jail time of 10 years. 3. If both of the suspects confess to having a gun, then they are both charged with the armed robbery but the jail terms will be negotiated down to 5 years in light of their confession.1 Let us also assume that for both suspects, the payoff from serving t years in jail is −t. Q: Define the game as we defined the grade game last time What are the three elements of any game? Player 1 1 Confess Hold Out Player 2 Confess Hold Out -5, -5 0, -10 -10, 0 -2, -2 This setting of the prisoners’ dilemma is adopted from Acemoglu, Laibson, List Microeconomics 2ed. 1 ECON3308.01: Game Theory in Economics Summer 2021 When we set up a game using the payoff table or matrix, we call this normal form representation of the game. Q: Recall what the objective of any game is? Each player wants to maximize their own payoff. But do we believe that people are actually good at pursuing this objective? The game of chess is a strategic game with a well-defined structure, but do we actually expect people to know how to maximize their payoff flawlessly in such a complex setting? When game theorists assume rationality, they assume that people are perfect at calculating the best strategy and at following it throughout the course of play. Assumption 1. Rationality is complete knowledge of one’s own interests and ability to perfectly calculate what actions will best serve those interests. Let us now solve the game. ‘Solving’ a game simply means predicting the outcome or outcomes of the game, assuming that the players are rational. Before we do that, let me introduce the concept of best response. A strategy si of player i is a best response to the strategies of the others s−i in the game if, taking the other players’ strategies as given, it gives player i at least as good of a payoff than any other strategy s0i available, or ui (si , s−i ) ≥ ui (s0i , s−i ) for all s0i Q: Let us now find the best response in the prisoners’ dilemma game. Write out all the best responses explicitly. Player 1 Confess Hold Out Player 2 Confess Hold Out -5, -5+ 0, -10− -10, 0| -2, -2 • Player 1’s best response to Player 2 choosing Confess is Confess. • Player 1’s best response to Player 2 choosing Hold Out is Confess. • Player 2’s best response to Player 1 choosing Confess is Confess. • Player 2’s best response to Player 1 choosing Hold Out is Confess. A pair of strategies is said to constitute a Nash equilibrium of the game if the two strategies are best responses to each other, or if no player has an incentive to deviate given what the other players are doing. Formally, a strategy profile (s∗i , s∗−i ) is an equilibrium if and only if ui (s∗i , s∗−i ) ≥ ui (s0i , s∗−i ) for all i, s0i 2 ECON3308.01: Game Theory in Economics Summer 2021 Therefore, we predict the equilibrium of the game to be (Confess, Confess) and the equilibrium outcome to be (−5, −5). Let us examine once again the two equilibrium strategies. We can notice that Confess is a best response for both the players no matter what the other player does. Q: How do we call such strategies? Such strategies are called dominant strategies. Last time we briefly talked about what strict and weak domination is. Now we are able to restate the definitions in a more formal way. A strategy is called dominant if it is always a best response. A strategy is called dominated if there exists a strategy that strictly dominates it. The difference between strict and weak domination will turn out to be important later on, so let me formally emphasize it. A strategy si of player i strictly dominates player i’s strategy s0i if, given the strategies of other players s−i , ui (si , s−i ) > ui (s0i , s−i ) for all s−i A strategy si of player i weakly dominates player is strategy s0i if, given the strategies of other players s−i , ui (si , s−i ) ≥ ui (s0i , s−i ) for all s−i ui (si , s−i ) > ui (s0i , s−i ) for some s−i Let’s go back to our prisoners’ dilemma game. In this case, (Confess, Confess) is a dominant strategy Nash equilibrium (DSNE) because all of the players in the game follow their dominant strategies. Throughout the class we will get familiar with different types of equilibrium concepts that will be applied to different types of games we will encounter. Dominant strategy Nash equilibrium is the strongest equilibrium concept, i.e. the hardest one to argue against. 2 Iterated Elimination of Dominated Strategies So far we have only considered examples of the 2x2 games. This means that if there is a dominant strategy, the other strategy necessarily has to be dominated. Let us consider a different 3x3 example. As you can see, there are no strictly dominated strategies for Player 1. For Player 2, Left strictly dominates Middle and Right. Thus these two strategies may be eliminated, leaving only Left. With only Left remaining, for Player 1, Straight strictly dominates both Up and Down, so they are eliminated, making the Nash equilibrium (Straight, Left). 3 ECON3308.01: Game Theory in Economics Player 1 Up Straight Down Summer 2021 Left 1, 5 2, 4 1, 5 Player 2 Middle Right 2, 4 5, 1 4, 2 3, 3 3, 3 3, 3 The process we just went through is called Iterated Elimination of Dominated Strategies. Removing dominated strategies reduces the size of the game, and then the “new” game may have another dominated strategy either for the same player or their opponent, which can once again be removed. This process of reduction will continue until no further reductions can be made. If this process ends in a unique outcome, then the game is said to be dominance-solvable. 4
