MobLab's most difficult mixed-strategy game, based on O’Neill (1987)1.
Players are paired, with one assigned the role of the Row player and the other the Column player. Each player has 4 cards: three number cards (2, 3, and 4) and the Joker. Each round, each player chooses one of the four cards. The cards are revealed, and the loser pays the winner five cents. The winner is determined by which of the following four mutually exclusive (and collectively exhaustive) events occurred:
Two Jokers: Row wins, receives 5 cents from column.
One Joker: Column wins, receives 5 cents from row.
Two Matching Number Cards: Column wins, receives 5 cents from row.
Two Different Number Cards: Row wins, receives 5 cents from column.
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. By default, each pair plays 5 rounds.
In the mixed-strategy equilibrium, each player chooses Joker with a 40% probability, and each of the number cards with a 20% probability. Unequal probability weights make this game more challenging than games like Rock, Paper, Scissors.
Note that Row’s equilibrium payoff is -1 in expectation. As this is a zero-sum game, Column’s is +1.
MobLab’s Matching Pennies and Rock, Paper, Scissors games are less complicated games where games where the only Nash equilibrium is in mixed strategies. While you cannot change the payoffs in the MinMax game, you can specify your own N×M matrix using MobLab’s Matrix: Instructor Sprcified 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, 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 of Row winning 40% of all games. Given that there are 16 potential outcomes (and all played in equilibrium), you will likely see significant between-round variation in outcome frequency, especially when the number of groups is small. (In equilibrium, the expected probability of Joker/Joker is 16%, of any of the six outcomes involving one Joker and one number cards is 8%, and any of the nine outcomes involving two number cards is 4%.)
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. Your students should come close to the equilibrium prediction, but you may see significant between-round variability when the number of groups is small.