It is 1850 during the days of the Gold Rush in California, and 100 miners are si

Question

It is 1850 during the days of the Gold Rush in California, and 100 miners are simultaneously which of three plots to go to and mine for gold. In order to simplify mattes, suppose it is known how much gold is in the ground at each plot: 1.000 ounces in plot A, 600 ounces in plot B, and 400 ounces in plot C. A miner^'s payoff is the number of ounces he mines, which is assumed to equal the number of ounces at the plot he^'s chosen divided by the total number of miners at that plot; thus, miners at a plot are assumed to work equally effectively, so that they end up with equal shares of the amount of available gold. Find a Nash equilibrium.

Explanation / Answer

Nash equlibrium is a mathematical 'description' of a stalemate, to be specific in terms of applicability and real meaning. That is, given n number of players in a situation of equal opportunity, none of the players will be able to 'win' if they were to adopt a certain strategy becasue Nash equlibrium assumes each player to know his rival's strategy. The result is a stalemate because every player thwarts every other player's attempts to win. So if player A and B were to play a game of chess and if there was to be a Nash equlibrium, then player A would be fully aware of B's strategy and B wuld be fully aware of A's strategy and nobody would win the game.

Now, keeping this in mind, say the miners choose plot A and exhaust all the gold there, then each miner will be able to dig 10 ounces of gold (1000/10). From that point of view, this is certainly a Nash equlibrium, since every miner of the group of 100 will be able to dig only 10 ounces of gold, which is equal for every miner at plot A.

Suppose plot A goes empty and the miners next choose plot B that has a total of 600 ounces of gold. Nash equlibrium assumes that each player will implement the best possible strategy to gain the maximum pay-off. At plot B, each miner will thus work the hardest to dig 6 ounces of gold (600/100). Nash equlibrium is achieved at plot B too.

The same applies at plot C that will yield 4 ounces of gold for each miner. So in essence, even if the number of miners were to get reduced to miner A and B, then both would yield 200 ounces of gold, of course assuming that each miner works at full potential.

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