x 0.3 Assignment 3: Describing Real Life Games Describing games When you describe a game there are two parts. Formal description Formally you need to present all the information that characterizes the game. For a game of independent or simultaneous choice you need to present the game in strategic form: The player set, the strategy space and the payoff function. These are all neatly summarized in the matrix form. For a game where decisions are made one after another, you also need to describe the order of play. Adding the order of play means you have a game in extensive form. It is usually convenient to draw the game tree with all the payoffs. For games of incomplete information you can show unknown payoffs with ”XX”. If some player does not know all the branches you might have to draw a separate version for each player. For games with imperfect information (where people don’t know what decision node they are at) you can draw a loop around the nodes the player cannot distinguish. You could connect the nodes with a dashed line instead. These connected nodes make up an “information set”. (The player knows they are in the information set, but not where in the information set.) Informal description The informal description is usually a story – describing a situation with real people. For example Matching pennies is the name for a simple example game used in game theory. It is the two-strategy equivalent of Rock, Paper, Scissors. (Note: Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.) 0.3. ASSIGNMENT 3: DESCRIBING REAL LIFE GAMES xi Here is a formal description of Matching Pennies The game can be written in strategic form using a payoff matrix like the one below. Each cell of the matrix shows the two players’ payoffs, with Player A’s payoffs listed first. Table 8: Matching Pennies Heads Tails Heads +1, -1 -1, +1 Tails -1, +1 +1, -1 Here is an informal description of the play of Matching Pennies The game is played between two players, Player A and Player B. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails) Player A keeps both pennies, so wins one from Player B (+1 for A, -1 for B). If the pennies do not match (one heads and one tails) Player B keeps both pennies, so receives one from Player A (-1 for A, +1 for B). This is an example of a zero-sum game, where one player’s gain is exactly equal to the other player’s loss. The Assignment NOTE: you may collaborate with members of your team and submit the same answers for Questions 2 and 4. For the other questions you must present individual answers, but of course you may discuss solutions. 1. Describe a situation that you encounter in your own life that can be represented as a game of complete information 2. Describe a second, and different situation that you encounter in your own life that can be represented as a game of incomplete information 3. Describe a situation that you encounter in your own life that can be represented as a game of perfect information 4. Describe a second, and different situation that you encounter in your own life that can be represented as a game of imperfect information xii 5. Solve the following centipede game. S stands for Stop and C stands for continue. there are two players, 1 and 2, and the player that chooses at each note in S S S S S S 1 C 2 C 1 C 2 C 1 C 2 C 1, 0 0, 2 3, 1 2, 4 5, 3 4, 6 6, 5 6. In the preceding game what do you think of the solution you found? Can you think of anything to improve the result? 7. What is a centipede game? Why is a centipede game interesting? 8. Replace the Xs in the following centipede game with payoffs that will induce a solution at the 5th exit (Player 1 will choose S at the 5th leg of the centipede). S S S S S S 1 C 2 C 1 C 2 C 1 C 2 C X, X X, X X, X X, X 5, 5 X, X X, X