MobLab
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# Rock, Paper, Scissors

## Game Description

The universal nature of this childhood game makes it ideal for demonstrating zero-sum games and mixed-strategy equilibria in a context slightly more complicated than Matching Pennies.

Each round, each of two players simultaneously chooses between Rock, Paper and Scissors. The loser pays the winner a penny. If they choose the same, the game is a tie (and no pennies change hands). For other combinations:

{Rock; Paper} Paper (covering rock) wins, receiving a penny from rock.
{Paper; Scissors} Scissors (cutting paper) wins, receiving a penny from paper.
{Scissors; Rock} Rock (crushing scissors) wins, receiving a penny from scissors.

### Learning Objective 1: Pure-strategy vs. Mixed-strategy Equilibria

While all games with a finite number of actions have a Nash equilibrium, not all games have an equilibrium in pure strategies. These games will have a mixed-strategy equilibrium.

### Learning objective 2: Mixed-strategy equilibrium

In a mixed-strategy equilibrium, each player chooses the action mixture making the other player indifferent between the actions she (probabilistically) plays in equilibrium. Indifferent between these actions, a player is willing to play the mixture necessary to make the other indifferent.

## Brief Instructions

The student’s game screen is shown in Figure 1. In the mixed-strategy equilibrium, each player chooses with a ⅓ chance, which causes her opponent to have the same expected payoff (0) from each of his actions. By default, each pair plays 5 rounds.

## Key Treatment Variations

MobLab’s Matching Pennies game is a slightly less complicated mixed-strategy equilibrium game. While you cannot change the payoffs in either of these games, you can specify your own N×M matrix using MobLab’s Matrix: Instructor Specified game.

## Results

At the top of the results page, we depict the payoff matrix (Figure 2). This may be helpful in establishing the lack of mutual best response in pure strategies: whenever row is best responding, column’s best response is to change his action.

Subsequent figures show how close your students are to equilibrium play. First, we display the frequency of each of the four outcomes for each round (Figure 3, revealed by clicking the triangle next to Outcome Frequencies). Your students should come close to the equilibrium prediction (each outcome occurring with a 1/9 probability, approximately 11%), but you may see significant between-round variability when the number of groups is small.

Next, we display the frequency of each action for each round (Figure 4, revealed by clicking the triangle next to Action Frequencies). You can reveal or hide the results for an action by clicking entry in the legend. Once again, your students should come close to the equilibrium prediction (each action occurring with a ⅓ probability), but you may see significant between-round variability when the number of groups is small.