We will loosely model the petrol crisis the UK has recently experienced as a game. Start with a two-player case. Assume that there are two people living on an island and each person has two actions: keep buying petrol as normal or rush to the petrol station and fill up the tank. When both players buy as normal, the petrol supply would be enough for the two, and hence no single player has any incentives to rush. In fact, when all players keep buying as normal, rushing unilaterally will incur a small cost (e.g. one has to give up their normal routine, go to the gas station with half full tanks etc.). If one player rushes, however, the other has incentives to do so too, for waiting when the other player rushes results in the worst outcome for the player who waits. Create a game using this scenario assuming that players decide simultaneously. Assign payoffs for all possible strategy combinations, carefully arguing and justifying why you assigned a particular payoff to a particular strategy combination. Any reasonable payoffs are allowed, provided that the game is within the general framework given above and the payoffs are justified well. Find all Nash equilibria of the game you created. Now assume that players play this game indefinitely repeatedly so that after every round the probability that the game is repeated is w with 0
We will loosely model the petrol crisis the UK has recently experienced as a game. Start with a two-player case. Assume that there are two people living on an island and each person has two actions: keep buying petrol as normal or rush to the petrol station and fill up the tank. When both players buy as normal, the petrol supply would be enough for the two, and hence no single player has any incentives to rush. In fact, when all players keep buying as normal, rushing unilaterally will incur a small cost (e.g. one has to give up their normal routine, go to the gas station with half full tanks etc.). If one player rushes, however, the other has incentives to do so too, for waiting when the other player rushes results in the worst outcome for the player who waits. Create a game using this scenario assuming that players decide simultaneously. Assign payoffs for all possible strategy combinations, carefully arguing and justifying why you assigned a particular payoff to a particular strategy combination. Any reasonable payoffs are allowed, provided that the game is within the general framework given above and the payoffs are justified well. Find all Nash equilibria of the game you created. Now assume that players play this game indefinitely repeatedly so that after every round the probability that the game is repeated is w with 0
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.8P
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Transcribed Image Text:We will loosely model the petrol crisis the UK has recently experienced as a game. Start with a two-player
case. Assume that there are two people living on an island and each person has two actions: keep buying petrol as normal or
rush to the petrol station and fill up the tank. When both players buy as normal, the petrol supply would be enough for the
two, and hence no single player has any incentives to rush. In fact, when all players keep buying as normal, rushing
unilaterally will incur a small cost (e.g. one has to give up their normal routine, go to the gas station with half full tanks etc.).
If one player rushes, however, the other has incentives to do so too, for waiting when the other player rushes results in the
worst outcome for the player who waits.
Create a game using this scenario assuming that players decide simultaneously. Assign payoffs for all possible
strategy combinations, carefully arguing and justifying why you assigned a particular payoff to a particular strategy
combination. Any reasonable payoffs are allowed, provided that the game is within the general framework given
above and the payoffs are justified well. Find all Nash equilibria of the game you created.
а)
Now assume that players play this game indefinitely repeatedly so that after every round the probability that the game
is repeated is w with 0<w< 1. Consider players can play the following strategies: All normal (try buying as normal
in all rounds), all rush (rush in all rounds), and gas-for-gas (start by buying as normal and keep buying as normal as
long as the other player buys as normal, rush if other player rushes in the previous round, and imitate the other
player's decision in previous round). Find the Nash Equilibrium/Equilibria of this game (possibly conditional on the
value of w).
Go back to the non-repeated version of the game above in a). Now assume that the game is played in sequence (e.g.
first player 1 acts, and then player 2). Find all Nash Equilibria and Subgame Perfect Nash Equilibria of this game.
Does the result differ when you assumed sequential play compared with the simultaneous play? If so in which way?
If not, why do you think sequence does not make a difference in this case?
Go back to the non-repeated simultaneous version of the game above in a). Now assume that there are 3 players
involved in the game, specify suitable payoffs for this version. Analyse the game with these 3 players: what is/are
the equilibria? Now assume that there are 4 players in the game, again specifying suitable payoffs for this extension,
what is/are the equilibrium in this version. How does the nature of the equilibrium depend on N-the number of
players in the game?
b)
c)
d)
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