Perhaps the simplest game to demonstrate zero-sum games and mixed-strategy equilibria.
Each round, each of two players simultaneously chooses between placing a penny heads up and placing it tails up. If the pennies match, the row player keeps both pennies. If the pennies do not match, the column player keeps both pennies.
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.
In a mixed-strategy equilibrium, each player chooses the action mixture making the other player indifferent between 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.
The student’s game screen is shown in Figure 1. In the mixed-strategy equilibrium, each player chooses heads with a 50% probability, which causes her opponent to have have the same expected payoff (0) from each of his actions. By default, each pair plays 5 rounds.
MobLab’s Rock, Paper, Scissors game is a slightly more 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.
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 (by matching column’s choice), 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 25% probability), but you may see significant between-round variability when the number of groups is small.
Next, we display the frequency of each action by player type for each round (Figure 4, revealed by clicking the triangle next to Action Frequencies). Once again, your students should come close to the equilibrium prediction (each action occurring with a 50% probability), but you may see significant between-round variability when the number of groups is small.