Question

The National Rock-Paper-Scissors League is about to have its annual com- petition and the CEO of Dunder Mifflin Paper Company thinks that this is a great advertising opportunity. She wants to sponsor the competition by offering an extra prize of x dollars (yet to be specified) to both players whenever both end up playing paper. Now when it came to budgeting, a heated discussion starts among employees about how offering this prize will affect the game and how often they will have to pay it. By the standard rules of the League, the loser must pay $10 to the winner of the game. If there is a tie (both players chose the same hand) then no payment is made. 1. (10 points) Write the normal-form game corresponding to the standard game of Rock-Paper-Scissors per traditional League rules (before the extra prize is introduced). Show that each player playing every strategy with equal probability is a Nash equilibrium. 2. (10 points) Consider Dunder Mifflin's introduction of a prize of 0 < x < 3 dollars to the game. Find a Nash equilibrium of this modified game (strategies may depend on x). In this equilibrium, what is the probability that in a given game, Dunder Mifflin will end up paying the extra prize?

Answer 1

About game:

The United States of America Rock Paper Scissors League was a national competition league for the hand game rock paper scissors.

Nash Equilibrium: Nash equilibrium is a concept within game theory where the optimal outcome of a game is where there is no incentive to deviate from their initial strategy. More specifically, the Nash equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice.

a) Equal probablity in Nash Equilibrium: Technically, a Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change his or her strategy given what the other players are doing.

We can model this situation by the following two-player strategic game.

Player 2
I   A
Player 1   I  
2,1   0,0
0,0   1,2
A
To find the Nash equilibria, we examine each action profile in turn.

(I,I)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.

(I,A)

By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing I rather than A.]

(A,I)

By choosing I rather than A, player 1 obtains a payoff of 2 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing A rather than I.]

(A,A)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.

We conclude that the game has two Nash equilibria, (I,I) and (A,A).

b)

(T,L)

Neither player can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.

(T,R)

Player 1 can increase her payoff from 0 to 1 by choosing the action B rather than the action T. Thus this action profile is not a Nash equilibrium.

(B,L)

Firm 1 can increase its payoff from 0 to 2 by choosing the action T rather than the action B. Thus this action profile is not a Nash equilibrium.

(B,R)

Neither firm can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.

We conclude that the game has two Nash equilibria, (T,L) and (B,R).