Economics 482 Game Theory and Economics - Problem Set 1 : Rock-Paper-Scissors and Dominant Strategy
Economics 482 Spring 2021
Due Tuesday, February 2
Problem Set 1
1. The game Rock-Paper-Scissors is popular in several countries around the world (some- times under a different name). It is a game for two players, in which the players take single actions simultaneously. Thus, it can be modeled as a simultaneous-move game. Rock-Paper-Scissors works as follows. On the count of three, each player announces one of three things: rock, paper, or scissors (usually using hand symbols rather than speaking aloud, but for the purposes of modeling the game, it does not matter). If the two players choose the same thing (for example, both choose scissors), then the result of the game is a tie. If the two players choose different things, then one player wins and the other loses, according to the following rule: a player who chooses rock beats a player who chooses scissors; a player who chooses scissors beats a player who chooses paper; and a player who chooses paper beats a player who chooses rock.
(a) How many outcomes does Rock-Paper-Scissors have?
(b) To define Rock-Paper-Scissors as a formal game, it is necessary to convert the notion of winning, losing and tying into a payoff function. Assume the following function: if a player wins, she gets a payoff of 1; if she ties, she gets a payoff of 0, and if she loses, she gets a payoff of -1. Using this payoff function, write down a matrix that represents Rock-Paper-Scissors. You can check your work against page 52 in the textbook.
(c) Let the players of Rock-Paper-Scissors be named Player 1 and Player 2. Suppose player 1 uses a mixed strategy in which σ1(R) = 1/2, σ1(P) = 1/4, and σ1(S) = 1/4, and player 2 uses a mixed strategy in which σ2(R) = 1/6, σ2(P) = 1/3, and σ2(S) = 1/2. Calculate each player’s expected payoff from this strategy profile.
(d) A friend of yours makes the following claim: “If you think that your opponent in Rock-Paper-Scissors is most likely to play rock, then you should play paper.” Decide whether you agree with your friend or not by performing the following exer- cise. Suppose that you are player 1, and that player 2 will use a strategy that puts more probability on rock than on paper or on scissors, i.e., 2’s mixed strat- egy (σ2(R), σ2(P ), σ2(S)) satisfies the properties σ2(R) > σ2(P ) and σ2(R) > σ2(S). Is it true that paper gives you a greater expected payoff than scissors or rock for any such strategy for player 2? If so, prove it. Or, can you find a strategy for player 2 satisfying this condition such that either scissors or rock (not necessarily both) would give you a greater payoff than paper? If so, identify one such strategy for player 2.
(e) Does either player have a dominant action in Rock-Paper-Scissors? Prove your answer.
2. A game with two players, Player 1 and Player 2, is represented in the matrix below. Player 1 has two possible actions, U and D, and Player 2 has three possible actions, L, M and R. Remember that when the payoff from an outcome is written in the form x,y, it means that x is Player 1’s payoff from that outcome, and y is Player 2’s payoff from that outcome.
Player 2
L M R
Player U 10,3 4,2 1,7
1 D 7,5 6,7 5,4
1. (a) Write down Player 1’s preferences over all of the outcomes of the game (that is, the order in which Player 1 likes the outcomes), noting any ties. Do the same for Player 2.
2. (b) Does player 1 have a dominant, or weakly dominant, action? Prove your answer.
3. (c) One of player 2’s actions is strictly dominated. Identify which action it is, and find a strategy (possibly mixed) for player 2 that strictly dominates the action you identify.
