To illustrate how game theory works, I sometimes pose this puzzle in class, taken from John McMillan's terrific book, Games, Strategies, and Managers.
QUESTION: Two pigs, one dominant and the other subordinate, are put in a pen. There is a lever at one end of the pen which, when pressed, dispenses 6 units of food at the opposite end. It "costs" a pig 1 unit of food to travel from the food to the lever and back.
If only one pig presses the lever, the pig that presses the lever must run to the food; by the time it gets there, the other pig has eaten 4 of the 6 units. The dominant pig can push the subordinate pig away from the food, and cannot be moved away from the food by the subordinate pig.
If both pigs press the lever, the subordinate pig is faster, and eats 2 of the units before the dominant pig pushes it away.
QUESTION: If each pig plays rationally, optimally, and selfishly, which pig will press the lever?
To answer the question, construct a simultaneous game, where the "payoffs" to the pigs are the net amount of grain consumed.
ANSWER: The subordinate pig always does better by not pulling the lever, regardless of what the the other pig does. This is called a "dominant strategy." The dominant pig's best response to this strategy is to pull the lever. The unique equilibrium is for the dominant pig to pull the lever and consume 1 net unit of food while the subordinate pig consumes four.
Subordinate pigQUESTION: Two pigs, one dominant and the other subordinate, are put in a pen. There is a lever at one end of the pen which, when pressed, dispenses 6 units of food at the opposite end. It "costs" a pig 1 unit of food to travel from the food to the lever and back.
If only one pig presses the lever, the pig that presses the lever must run to the food; by the time it gets there, the other pig has eaten 4 of the 6 units. The dominant pig can push the subordinate pig away from the food, and cannot be moved away from the food by the subordinate pig.
If both pigs press the lever, the subordinate pig is faster, and eats 2 of the units before the dominant pig pushes it away.
QUESTION: If each pig plays rationally, optimally, and selfishly, which pig will press the lever?
To answer the question, construct a simultaneous game, where the "payoffs" to the pigs are the net amount of grain consumed.
ANSWER: The subordinate pig always does better by not pulling the lever, regardless of what the the other pig does. This is called a "dominant strategy." The dominant pig's best response to this strategy is to pull the lever. The unique equilibrium is for the dominant pig to pull the lever and consume 1 net unit of food while the subordinate pig consumes four.
Pull Don't Pull
Pull (3,1) (1,4)
Dominate pig
Don't Pull (6,-1) (0,0)
Ironically, with these payoffs, the subordinate pig will soon become dominant. Then the equilibrium will change.
Why will the equilibrium change?
ReplyDeleteThe Nash point will remain the same, but the identities of the players will flip.
The equilibrium will change for the pigs. This is true.
ReplyDeleteDoes this game take into account jealousy? Maybe the pig will refuse to pull the lever because the other pig gets more. I know that will be a costly play by the pig but Nash does not take in account jealousy or punishment. Nash only describes making the best move for yourself.
I guess Nash equilibrium considers players to be rational, acting optimally, which avoids the question of jealousy.
ReplyDeleteBut I would think the chances of players figuring out the best group outcome (non-Nash outcome) will be more as both players act rationally.
ReplyDeleteAnother interesting point is that when Pig A goes from being subordinate to dominant and doesn't hit the lever for the first time, Pig B will become dominant again after that feeding, then Pig A will hit the lever and become dominant after that feeding, on and on. That's a net gain of 5 food for each pig per every two feedings, which is a better overall outcome for both than any of the other scenarios.
ReplyDelete