EC 308 Lecture Fill in the Blanks_Game_Theory_and_Auctions_Solutions_Fall_2023

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EC 308 Lecture Fill in the Blanks – Game Theory 1. What is the definition of a ‘strategy’ in game theory? A strategy is a complete contingent plan of action that specifies what action a player will take in every situation the player could come to in the game. 2. What is a dominant strategy? A dominant strategy is a strategy which yields a higher payoff to a player than any of the player’s other strategies, regardless of what the other players do. 3. What is the key property of a Nash equilibrium? The key property of a Nash equilibrium is that no player can benefit from a unilateral deviation. If one player could get a higher payoff by choosing a different strategy, holding everyone else’s strategy fixed at strategy profile S, then S is not a Nash equilibrium. In other words, if you fix everyone’s actions at a strategy profile, S*, then S* is a Nash equilibrium if no player can benefit from doing something else, keeping everyone else’s strategy fixed at S*. 4. Does either player have a dominant strategy in the “Nash Example” game? Explain why or why not. Nash Example Game Left Right Up 60,40 70, 60 Down 80,50 10, 30 Neither player has a dominant strategy. The Row player does better off playing Up if the Column player plays Right, but the Row player does better off playing Down if the column player plays Left. Hence, the Row player’s best -response strategy depends on what the other player chooses. Similarly, the column player’s best response depends on what the row player does.

5. What are the Nash equilibria for the “Nash Example” game? Explain why. The Nash equilibria are {Down, Left} and {Up, Right}. If the row player plays Down and the column player plays Left, then the row player gains 80 and the column player gains 50. If the row player instead deviated to Up, the row player would earn 60 instead of 80. If the column player instead deviated to Right, the column player would earn 30 instead of 50. So neither player can benefit from unilaterally deviating, and {Down, Left} is a Nash equilibrium. Similarly, if the row player plays Up and the column player plays Right, then the row player gains 70 and the column player gains 60. If the row player instead deviated to Down, the row player would earn 10 instead of 70. If the column player instead deviated to Right, the column player would earn 40 instead of 60. So neither player can benefit from unilaterally deviating, and {Up, Right} is a Nash equilibrium. 6. What are two examples of games with only a mixed strategy Nash equilibrium? Some examples of games with a unique mixed strategy equilibrium are:  Rock-paper-scissors  Soccer  Hockey  Tennis In these games, players must randomize their strategies to surprise their opponent. If their strategies are too predictable, they will be giving their opponent an advantage. 7. What is a coordination game? What is the key property of a coordination game? A coordination game is a game with multiple Nash equilibria. It is called a coordination game since players need to figure out how to coordinate on one particular equilibrium. The key property of a coordination game is that it has multiple Nash equilibria. 8. What is an auction? An auction is a set of rules for determining: 1) How to allocate scarce resources 2) Who should pay for those resources 3) How much should be paid for those resources It is essentially a market where the rules of the market are designed in advance.

9. What is the optimal bidding strategy in a second price sealed bid auction? The optimal bidding strategy in a second price sealed bid auction is to bid exactly how much you feel the item is worth to you (to bid your value for the item). This strategy is optimal since:  If you bid more than your value and win and the second highest bidder also bids more than your value, than you pay more than the item is worth to you.  If you bid less than your value and lose and the highest bidder also bids less than your value, than you might have won the auction and earned a positive surplus if you had bid your value.  Bidding exactly your value in a second price sealed bid auction ensures that you win the auction exactly when you want to win it and lose the auction exactly when you want to lose it. 10. What is the relationship between a second price sealed bid auction and an ascending auction? A second price sealed bid auction and an ascending auction are strategically equivalent. In both auctions it is optimal to bid exactly your value. (In the ascending auction, this means dropping out of the auction when the price rises above your value). 11. In a first price sealed bid auction is it optimal to bid your valuation for the object at auction, to bid above your valuation, or to bid below your valuation? Explain. In a first price sealed bid auction it is optimal to bid below your value. If you bid above your value and win the auction, you pay what you bid and so you earn a negative surplus. If you bid exactly your value and win the auction, you pay what you bid and earn a zero surplus. The only way to win a positive surplus in a first price sealed bid auction is to bid below your value. But if you bid too low, you are more likely to lose the auction. So there is a tradeoff: Bidding higher increases the probability you win the auction, but decreases your surplus if you win (in a first price sealed bid auction).

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