Sample Problem 11

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ECON 2420 Oligopoly Charles L. Baum II Judy wants to support her son, Matthew, if he looks for work but not otherwise. Matthew wants to try to find a job only if Judy will not support him. Their payoff matrix is illustrated below. a) What is Judy’s dominant strategy, if any? b) What is Matthew’s dominant strategy, if any? c) What is the Nash equilibrium? d) What is the Stackelberg equilibrium if Matthew must move first? Matthew (M) Look for Work Loaf Judy (J) Support M = 2 J = 4 M = 4 J = -1 No Support M = 1 J = -1 M = 0 J = 0 Judy picks the row and Matthew picks the column. A dominant strategy is a strategy that works better than any other strategy regardless of what the other player is doing. A Nash Equilibrium is where each player is doing the best that they can given what the other player is doing. A Stackelberg Equilibrium is the equilibrium to a sequential-move game. Judy does not have a dominant strategy. If Matthew looks for work, then she is better off supporting him (with a payoff of 4) than not supporting him (with a payoff of -1). However, if Matthew loafs, then Judy is better off not supporting him (with a payoff of 0) than supporting him (with a payoff of -1). Matthew does not have a dominant strategy. If Judy supports him, then he is better off loafing (with a payoff of 4) than looking for work (with a payoff of 2). However, if Judy is not supporting him, then he is better off looking for work (with a payoff of 1) than loafing (with a payoff of 0).

ECON 2420 Oligopoly Charles L. Baum II This game does not have a Nash equilibrium. For each outcome, at least one of the two players can increase their payoff by changing their behavior. If Judy supports Matthew and Matthew looks for work, then Matthew could increase his payoff by loafing. If Judy supports and Matthew loafs, then Judy can increase her payoff by not supporting. If Judy does not support and Matthew looks for work, then Judy can increase her payoff by supporting, and if Judy does not support and Matthew loafs, then he can increase his payoff by looking for work. If Matthew moves first and looks for work, then Judy will maximize her payoff by supporting and Matthew’s payoff will be 2. If Matthew moves first and does not look for work, then Judy will not support and Matthew’s payoff will be 0. So Matthew maximizes his payoff by looking for work, and Judy consequently supports. This is the Stackelberg equilibrium.

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