Northern Virginia Community College Economics Game Theory Questions

1. Draw a game tree with three players who choose in sequence; where the 1stplayer has four actions(a, b, c, d); the 2ndplayer also has four actions (w, x, y, z); and the 3rdplayer has two actions (u, v).Player 2 knows when player 1 chose action b or c, but can’t distinguish which of the other twoactions were chosen by player 1. Player 3 knows when player 1 chose action a or c, but can’tdistinguish which of the other two actions chosen by player 1. Player 3 knows when player 2 choseaction x or y, but cannot distinguish which of the other two actions were chosen by player

2. Drawthe information sets for this case.2. Start with the original matching pennies game, where player 1 wins by matchups and player 2 winsby mismatches; and the winning versus losing payoffs are +1 and – 1 respectively. Then decreasethe payoff from a mismatch loss for player 1 (when players 1 and 2 choose tails and headsrespectively) from–1 down to – 10, and decrease the losing payoff from a double tails loss for player2 from –1 down to – 8; but keep all other winning or losing payoffs the same as before (at either +1or – 1).Draw the payoff matrix with the modified losing payoffs. Also draw two diagrams showing eachplayer’s pair of expected payoff lines, and showing the other player’s equilibrium decisionprobability. Finally, calculate the numerical values ofthese Nash equilibriumprobabilities (p1*, p2*)for this case.

3. A. Draw the causal tree for the weather forecasting example, and then use it to derive formulas forthe conditional probabilities p[R*fb] and p[R*sb]; where ë is the probability of histories that willcause rain instead ofsunshine later, and á versus â are the probabilities of a falling instead ofsteadybarometer caused by rain producing versus sunshine producing histories respectively.B. With the above formulas, show the algebra steps needed to prove the following theorem:p[R*fb] > p[R*sb] if and only if á > âC. Calculate a numerical example of the above theorem by calculating the numerical values of thetwo conditional probabilities p[R*fb] and p[R*sb], when (ë, á, â) = (.3, .7, .2).

4. A. Draw the causal network for Newcomb’s Problem with perfect detection and use it to explainwhy the following argument is valid or not; where the guaranteed money in the clear box is denotedby L and the larger amount of money that might be in the black box is denoted M.“Once a given amount of money is put into the black box, it is irreversibly fixed no matterhow many boxes Adam might open in the future.“So if Eve had put nothing in the black box, and if Adam later opens both boxes, then he willobtain $L compared to $0 if he instead opens only the black box. Likewise, if Eve had already put$M in the black box, and if Adam later opens both boxes, then he will obtain $M + $L compared toobtaining $M if he instead opens only the black box.”“Thus, regardless of what Eve has already done, Adam will obtain an additional $L if heopens both boxes instead of opening only the black box. So Adam is guaranteed to obtain moremoney by opening both boxes, which therefore must be his rational decision.”

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