Let’s say that there are three fishermen, and each day they individually decide how many boats to send out to catch fish in the local lake.
A fisherman can send out one or two boats, and the daily cost of a boat is $15. The more boats sent out, the more fish are caught. However, since there are only so many fish to be caught on a given day, the more boats another fisherman sends out, the fewer fish the remaining fishermen can catch. Things need to be shared, but not everyone wants to share. Overexploitation is very real.
The accompanying table reports the size of a fisherman’s catch, depending on how many boats each fisherman sends out.
A fisherman’s current-period payoff is the value of his catch (assume that each fish sells for a price of 1), less the cost of the boats. For example, if a fisherman sends out two boats and the other two fishermen each send out one boat, then a fisherman’s payoff is 75 – 30 = 45.
The stage game is symmetric, so the table is to be used to determine any fisherman’s payoff. The fishermen play an infinitely repeated game where the stage game has them simultaneously choose how many boats to send out. Each fisherman’s payoff is the present value of his payoff stream, where fisherman i’s discount factor is “S”.
- What would result in a payoff higher than that achieved at Nash equilibria for the stage game?
- What would be the grim-trigger strategy that results in the actions being implemented, and what are the conditions for that strategy to be a symmetric Subgame Perfect Nash Equilibrium?
Grim Trigger Strategy
A grim trigger strategy is the pareto-improving outcome, where we basically punish someone forever if they cheat, or stray from the equilibrium. You can operate an equilibrium forever, but if you stray from that equilibrium for a higher payoff today, you will be punished with a lower payoff for the rest of eternity (pretty intense).
Basically, there is a value to cooperating and there is a value to defecting. In order for this game to work, the value of cooperating must be greater than the value of defecting – people need to want to get along with each other. In order for people to get along and not completely betray the other, they must know that there is some probability that they will see this person tomorrow.
So for our fishermen, what is the probability that tomorrow will happen? We can determine that by calculating the advantage of cheating and the advantage of cooperating. The advantage of cheating in this case would be equal to:
So for our fishermen, the value of the reward today would be the highest payoff that they can get in this situation, in which everyone sends out two boats, with a payoff of 45. However, forever after, everyone will be stuck with a payoff of 20.
The advantage of cheating is 20. So how does that compare to the advantage of cooperating? The advantage of cooperating is similar, but we take the value of the reward forever, which we discount at rate S minus the value of punishment forever (if we choose to cheat) also discounted at rate S, to represent forever.
We then would set the advantage of cheating as an inequality against the advantage of cooperating.
Solving that out, we get that the discount factor must be greater than or equal to 4/5 or 80% in order for this grim trigger to work. So basically, there must be an 80% chance of tomorrow occurring in order for the fishermen to stay at the grim trigger strategy, and not cheat.
No one likes the Tragedy of the Commons, and one way to avoid that is to make sure that people know that tomorrow will happen. Our fishermen will cooperate with each other if they know that there is an 80% chance that tomorrow will occur and they will see each other. Thinking of the future can keep people in check.
“Picture a pasture that is open to all. Each herdsman will try to keep as many cattle as possible on the commons…the inherent logic of the commons remorselessly generates tragedy.” —Garrett Hardin, The Tragedy of the Commons 1968