Game Theory in the Social Sciences

True or false, and explain.

It is not possible for a finite game to have a mixed strategy Nash equilibrium, if all players have strictly dominant strategies.

Consider a contributions game with 2 players. Each player can either ‘Contribute’ or ‘Not.’ If either (or both) Contribute, a good is provided to both. The good is worth 2 jollies to each player. If a player Contributes, she pays a cost of 1 jolly (deducted from the 2 jollies that the good is worth to her). If a player does Not contribute, she has no cost deducted from her jollies. If both players choose Not to contribute, both players receive 0 jollies.

(a) Suppose player 1 uses a mixed strategy. What is player 2’s expected payoff?

(b) Suppose player 2 uses a mixed strategy. What is player 1’s expected payoff?

(c) What is the mixed strategy Nash equilibrium in this game?

(d) Suppose the good is worth 3 jollies to both players instead of 2. What is the mixed strategy Nash equilibrium of this game?

Consider a 2 player stag hunt in which a Stag is worth 6 jollies to each player and a Hare is worth 1 jolly to any player catching one. What is the mixed strategy Nash equilibrium in this game?

. Pat and Chris must independently decide whether to go to Naan n’ Curry or Top Dog at noon. Pat prefers Naan n’ Curry; Chris prefers Top Dog; and as BFF’s, they derive no jollies unless they eat together. Specifically, if they both choose Top Dog, Chris earns 3 jollies and Pat 1 jolly, and vice versa if they eat at Naan n’ Curry. 1

(a) Draw the strategic form

(b) Identify all Nash equilibria

Pat and Chris have had a falling out and are now mortal enemies. They must independently decide where to eat, as in the previous question. Pat still prefers Naan n’ Curry; Chris still prefers Top Dog. Each player gets 1 jolly from eating at her less preferred place and 3 jollies from eating at her more preferred place. Now, they each lose 1 jolly deducted from these payoffs if they eat at the same place.

(a) Draw the strategic form

(b) Identify all Nash equilibriam