consider the goldminers during the Gold Rush. There are 250 miners who simultaneously decide which of...

consider the goldminers during the Gold Rush. There are 250 miners who simultaneously decide which of the three plots to go to and mine for gold. It is known to everyone that there will be 2,500 ounces in plot A by Anchorage, 1,500 ounces in plot B by Big Lake and 1000 ounces in plot C by Cold Bay. A miner’s payoff is the number of ounces at the plot he has chosen divided by the number of miners at that plot. Find the Nash equilibrium(ia).

Solutions

Expert Solution

Solution. At Nash equilibrium, no player can gain by changing his/her strategy given that all the other players' strategy remains unchanged. This means that all the players have played their best possible(giving him maximum gain possible) move in regards to the other players' moves and so any change would leave him worse off comparatively.

In our given situation the miners can go to any of the plots A, B or C. A miner from plot B (suppose) would leave to plot C or A if he gets more gold comparatively. Hence any miner would move to a new plot from an older plot if there is a prospect to higher quantities of gold. At Nash equilibrium no miner(player) changes their plot(position/strategy). Hence at Nash equilibrium, all the miners receive equal quantities of gold.

Total gold = 2500 + 1500 + 1000 = 5000
Total Miners = 250
Miners per ounce = 5000 / 250 = 200

Hence miners in plot A = 2500 / 200 = 125
Miners in plot B = 1500 / 200 = 75
Miners in plot C = 1000 / 200 = 50 Total Miners = 250

Now if any miners change his position from (say) plot B to C he would get less amount than he was getting at plot B due to overcrowding at plot C. Hence no miner would change his position and any change in their positions would leave them worse off.

Hence at Nash Equilibrium, the number of miners

Plot A = 125
Plot B = 75
Plot C = 50

Rahul Sunny answered 1 year ago

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