Case study

GAME 1 Rock-Scissors-Paper

This game entails playing three different versions of the children's game rock-scissors-paper. In rock-scissors-paper, two people simultaneously choose either Rock, Scissors, or Paper, usually by putting their hands into the shape of one of the three choices. The game is scored as follows: A per- son choosing Scissors beats a person choosing Paper (because scissors cut paper). A person choosing Paper beats a person choosing Rock (because paper covers rock). A person choos- ing Rock beats a person choosing Scissors (because rock breaks scissors). If two players choose the same object, they tie. Technically, players end up mixing over three possible pure strategies, and the analysis for this is not presented until Chapter 7. You can modify this in-class game so that it can be used to teach about two-by-two games, use it as is but focus your discussion on the points relevant to two-by-two games, or play it immediately before covering the analysis of larger games.

Instructions for students: In this experiment, each indi- vidual game (and we will play multiple games) is worth 10 points. Because it is a zero-sum game, the winner gets 10 points and the loser gives up 10 points. The following matrix shows the possible outcomes in the game.

In all three of the games described below, you are the row player and your professor is the column player. For each situation, we will simulate 60 repetitions of a game. Because actually signaling with our hands 60 times would be too time-consuming, you are instead asked to describe your behavior by writing down how many times (on average) out of 60 games you would play Rock, how many times you would play Scissors, and how many times you would play Paper.

Your professor will choose a particular strategy to use in each of three situations; her strategy in Situation 1 differs from her strategy in Situation 2, and so on. In each situation, your professor uses the same strategy against every student. You will be told your professor's exact strategies for Situa- tion 1 and Situation 3 when we analyze the outcomes of the games; your professor's exact strategy for Situation 2 is ex- plained below. Your task in each of the three games is to choose the strategy that you think (or guess) is most likely to maximize your total payoff.

SITUATION 1

This is the regular version of the game, using the payoffs above. You are asked to write down how many times (on average) out of 60 you would play Rock, how many times you would play Scissors, and how many times you would play Paper. In a similar way, your professor has written down numbers that describe her behavior. You can assume that your professor plays her equilibrium strategy here.

SITUATION 2

In this version of the game, the payoffs remain as above, but your professor commits (for some reason) to picking Rock 24 times out of 60 (40%), picking Scissors 18 times out of 60 (30%), and picking Paper 18 times out of 60 (30%). (The order in which she will make these picks is unknown.) No matter what anybody writes down, your professor's behavior will follow this rule. Knowing this, again try to maximize your total payoff.

SITUATION 3

In this version of the game, the row player (you) has an ad- vantage. If you pick Rock and your professor picks Scissors, you win 20 points from your professor. The payoff matrix is now:

This time, your professor will use a strategy that she thinks will be beneficial for her (but she does not know for certain that it will be beneficial). Again, try to maximize your total payoff.

When discussing the results of these games, you will be able to note that everyone's payoff in Situation 1 is identical, regardless of her choice. This is a direct result of the fact that equilibrium mixtures keep a rival indifferent; any pure strategy or combination of pure strategies yields the same payoff against someone playing her equilibrium mixture. In Situation 2, the professor does not play her equilibrium mix. Some students (but perhaps not many) will figure out that this is not the equilibrium mix and that it can therefore be exploited. You can show on the board during the discussion that the student, using pure Rock, can obtain an expected payoff of 0 per game; the student, using pure Scissors, can obtain an expected payoff of -1 per game (so -60 over 60 plays of the game); or the student, using pure Paper, can ob- tain an expected payoff of 1 per game and 60 over 60 plays of the game. Given the disequilibrium mixture, the student can take advantage of the professor's "error" by playing Rock 100% of the time.

Finally, in Situation 3, you can consider a number of dif- ferent strategies. If you again use your equilibrium mix, then the payoff to you is 5/6 (on average) per game, for a total of 50 over the 60 games. We have also used a number of nonequilib- rium mixes on occasion; the most interesting one to analyze is the one that avoids the use of Scissors entirely, putting equal weight on Rock and Paper. This helps to make the point about the counter-intuitive outcome-that the student wants to avoid Rock in the new equilibrium mix-even more stark. Against a professor mixing only Rock and Paper in equal proportions, the student does best to use all Paper.

This game can be used to motivate: (1) the calculation of equilibrium mixes for larger games; (2) that a rival's equi- librium mix keep you indifferent among your pure strategies or any mixture, so makes your choice of strategy irrelevant; (3) that exploitation is possible if a rival is using a nonequilib- rium mix-one of your pure strategies will be dominant; and (4) the counterintuitive outcome that, in equilibrium, you decrease your use of a strategy for which one payoff has increased.