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.
Sounds like a fun class! :-) Here are my wikipedia-trained (i.e not so trained) thoughts: If we model the situation as each player having two possible choices, the outcome seems fairly intuitive (but I could still be wrong). I predict that the big bad pig chooses to pull the lever and gets a payoff of 1 and the smaller pig waits by the food and gets a payoff of 4. However, if one models the situation as a choice between three options, where the additional option is to walk towards the lever only if the other pig is also doing so and otherwise wait, the game gets a lot trickier to predict. Then I would go for a mixed strategy, both pigs fully randomly selecting which option to pursue, but I am far from sure if that is the "rational" strategy. Speculating forward: As far as my intuition goes, introducing randomly limited iteration does not seem to mess up the predictions of the simple two-options-model, but might do it for the three-options-model. As for which model is the more suitable, I suppose it boils down to whether we believe the pigs can see each other in realtime or not.
ReplyDeletewhat's the answer?
ReplyDeleteSubordinate pig
ReplyDeletePull Don't Pull
Dominate pig Pull 2,2 1,4
Don't Pull 6,0 0,0
Dominate pig wants subordinate pig to pull, so he can eat all 6 units. Subordinate pig wants dominate pig to pull, so he can have 4 units before he's inevitably pushed out of the way.
That said, looks like they'll both be pulling. Pigs gotta eat.
To so confidently assume that there is only one correct way to model the situation (in the suggested answer with two available strategies, pull and !pull) and not to discuss the alternative models. Insincere or only naive? ;-)
ReplyDelete