Question

Suppose that Player 1 is a police officer who must decide whether to patrol the streets or hang out in the café. Their payoff for hanging out in the café is 10, while their payoff for patrolling the streets depends on whether they catch a thief, who is Player 2. If the thief is roaming the streets, the police officer will catch them and receive a payoff of 20. If the thief is hiding, the officer's payoff is 0. The thief must choose between staying hidden or roaming the streets. If they stay hidden, their payoff is 0, while if they roam the streets, their payoff is -10 if the officer is patrolling the streets and 10 if the officer is in the café. Write the matrix form of this game.

Answer 1

The payoff matrix for the game between a police officer and a thief with strategies to patrol or stay in a café for the officer and to roam or stay hidden for the thief, is depicted in a table. Each cell lists the payoffs for both players based on their chosen strategies.

Step-by-step explanation:

The student's question involves constructing the payoff matrix for a two-player game in the context of game theory. The game includes a police officer (Player 1) and a thief (Player 2), with each player having two strategies. For Player 1: Patrol Streets (P) or Hang in Café (C). For Player 2: Roam Streets (R) or Stay Hidden (H). The payoffs are given as outcomes based on the choices made by both players. The objective is to write the matrix form of this game.

To create the payoff matrix, we list Player 1's strategies as rows and Player 2's strategies as columns, filling in the outcomes for each strategy combination. Here's the matrix:
RH
P20, -100, 0
C10, 1010, 0

The numbers in each cell represent the payoffs for Player 1, Player 2 respectively. For instance, if Player 1 patrols (P) and Player 2 roams (R), Player 1's payoff is 20, and Player 2's payoff is -10.