Consider the following game where a robber is attempting to steal a diamond, and

Question

Consider the following game where a robber is attempting to steal a diamond, and a police officer is on duty to guard the diamond. The officer is bored however and considers going to get a doughnut. The normal form of the game is represented below where the officer is the column player and the robber is the row player.

A) Find any Nash equilibria of this game.

B) Suppose the officer notices the "hot doughnuts" sign is on, so the payoff of eating a doughnut when the robber does not steal increases from 2 to 4 (the rest of the payoffs are unchanged). How would this change the officer's decision, if at all? What is the new equilibrium?

Guard Doughnut Steal -5,5 10,-4 Not 0,-1 -1,2

Explanation / Answer

The Nash equilibrium is equilibrium strategy of a 2 player non-cooperative game where each player knows the equilibrium strategy of the other.

The game table is given in the figure below:

In the game above we consider the strategy of the thief at a time to see the response of the guard. Suppose, the theif decided to steal, then the officer has two strategy to play, he can either "GUARD" or "DOUGHNUT". In case of the thief play "STEAL" the offficer will play "GUARD" becuse this will give him the highest payoff 5.

Again, the theif decided to not steal, then the officer has two strategy to play, he can either "GUARD" or "DOUGHNUT". In case of the thief play "NOT" the offficer will play "DOUGHNUT" becuse this will give him the highest payoff 2.

Then there is two pure strategy nash equilibria; (STEAL,GUARD) with payoff [-5,5] and (NOT,DOUGHNUT) with payoff [-1,2]

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b. If the payoff of the officer doughnut strategy payoff in case the theif not steal increases to from 2 to 4, the game table altered as

In the game above we consider the strategy of the thief at a time to see the response of the guard. Suppose, the theif decided to steal, then the officer has two strategy to play, he can either "GUARD" or "DOUGHNUT". In case of the thief play "STEAL" the offficer will play "GUARD" becuse this will give him the highest payoff 5.

Again, the theif decided to not steal, then the officer has two strategy to play, he can either "GUARD" or "DOUGHNUT". In case of the thief play "NOT" the offficer will play "DOUGHNUT" becuse this will give him the highest payoff 4.

Then there is two pure strategy nash equilibria; (STEAL,GUARD) with payoff [-5,5] and (NOT,DOUGHNUT) with payoff [-1,4]

So, there will be no change in the outcome of the game.

Guard doughnut Steal -5,5 10,-4 Not 0,-1 -1,2

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