Global Warming is no Stag Hunt

Stag Hunt

Is finding a solution to Global Warming really like solving a Prisoner’s Dilemma?

Most real-world problems of cooperation and coordination, even if they are said to be one, are really not a Prisoner’s Dilemma but a Stag Hunt.

Knowing the difference is important to understanding which policies may work in practice, and under what conditions.

This post is technical in that it has some game theory in it. I will explain the necessary concepts in the spirit of seeking just enough simplicity without destroying their power and their beauty. You have been warned, so we jump right into it:

A Nash Equilibrium (named after John Nash who was the basis of the famous movie “A Beautiful Mind”) occurs when players of a non-cooperative game are each playing strategies where no-one has anything to gain by switching strategies unilaterally. One example is the decision on whether to drive on the left of the right side of the road (this is in fact a Coordination Game): If every-one drives on the same side, I have nothing to gain by driving on the “wrong” side. Everyone driving on the right, or everyone driving on the left, are both Nash Equilibria.

The Stag Hunt is also a Coordination Game with two Nash Equilibria, but they are not quite as symmetric as the driving laws are. In the literal version of the Stag Hunt, players choose to hunt a stag or a hare. There is no chance of catching a stag if you are the only one hunting it. The chances of catching a hare are what they are, independent of what others do, while the chances of catching a stag go up sharply with the number of stag hunters. The stag is much more valuable than the hare. So you see the situation is not as symmetric as the question “do we drive on the right or on the left?”. We are all better off cooperating and hunting stag, even though when everyone is hunting hare, we have no motivation to switch into a cooperative strategy all by ourselves. But it is much better for everyone if we all hunt the stag. So the question is under what conditions cooperation may occur in the Stag Hunt:

(1) The more players we need to pull off a successful stag hunt, the less likely cooperation is to occur

(2) Trust between the players helps, as do checks that players aren’t pretending to do one thing while doing another

(3) Knowing (or even wrongly believing) that the game will be played repeatedly with the same players will dramatically increase cooperation even if people don’t consciously think about the game or the outcomes: Even animals, and even bacteria exhibit cooperation under these same conditions.

To repeat: in the Stag Hunt, I have an incentive to do what I believe everyone else is doing. When everyone hunts stag and cooperates, it’s a Nash equilibrium, as I have nothing to gain from hunting hare instead, because the stag is much more valuable. Equally, when everyone is hunting hare, it’s a Nash Equilibrium (but one with a lesser outcome for all), as I have nothing to gain by hunting a stag on my own instead.

Stag Hunts are truly everywhere. As Brian Skyrms puts it:

In a larger sense, the whole problem of adopting or modifying the social contract for mutual benefit [i.e. moving a society from one Nash Equilibrium to a much better one] can be seen as a stag hunt. For a social contract to make sense, the state of nature [without the social contract] must be an equilibrium. Otherwise, there would not be the problem of transcending it. And the state where the social contract has been adopted must also be an equilibrium. Otherwise, the social contract would not be viable.

But instituting a social contract involves energy in setting it up, and — equally importantly — risk

The problem of instituting, or improving, the social contract can be thought of as the problem of moving from riskless hare hunt equilibrium to the risky but rewarding stag hunt equilibrium.

A Prisoner’s Dilemma, on the other hand, is a situation where cooperation from both “players” leads to the optimal overall outcome, but “defection” or “cheating” leads to superior outcomes for the one who cheats, regardless of what the others do. We say that, in a Prisoner’s Dilemma, defection is the dominant strategy, because it is always better for the player than cooperation, even if collective defection is the worst outcome for the whole community of players overall. For a Prisoner’s Dilemma to be present, the act of cheating has to be unconditionally beneficial to the individual (and therefore go largely unpunished even though the price to the community of players is high). While the stability of the cooperative equilibrium in the Stag Hunt is threatened only in the presence of doubts that others will cooperate, the stability of cooperation in the Prisoner’s Dilemma does not exist at all [1].

Let us take the cold-war arms race, which was (mistakenly) understood by most strategists as a Prisoner’s Dilemma: Both sides would be unable to attack if they have arsenals for mutually assured destruction (or MAD, an expensive Nash Equilibrium), and both sides would be unable to attack if they had no arsenal at all. The latter scenario was not attainable, because it was too easy to “cheat”, and almost impossible to be punished even if cheating were detected. So the outcome was inevitably MAD. But that doesn’t make the nuclear arms race a Prisoner’s Dilemma. It was a Stag Hunt, and each of the following two reasons would alone be sufficient to make it so: (A) Disarmament was both desirable from both sides and a Nash Equilibrium if the other side could be trusted to stick to disarmament [2], and (B) the threat of mutually assured destruction was not credible [3].

The last term we introduce is that of strict efficiency: a strategy is called strictly efficient if the population as a whole is better off all playing this strategy, than all players playing a different strategy (but all playing identically to each other). With our new terminology, we could have simply defined a Stag Hunt game as one with two (or multiple) Nash Equilibira, but one of these is better than all others (strictly efficient).

The purpose of policy, social contract, morals, and what have you is then to move populations from the one Nash Equilibrium (where, remember, no-one has an incentive to change) to another that is strictly efficient, and keep their behavior there.

This is very, very hard in practice. Consider for example a country where everyone cheats on taxes. That is not efficient, and yet it is very hard to get from such a state to one where everyone pays taxes and no-one has an incentive to cheat.

You have to create an environment where

(1) the intended behavior is strictly efficient,

(2) the intended behavior is a Nash Equilibrium (conditions (1) and (2) will turn the game from a Prisoner’s Dilemma to a Stag Hunt)

(3) persuade a majority of players all at once to leave the non-cooperative Nash Equilibrium, and

(4) give all players confidence that you have done point (3) successfully, i.e. a vast majority, if not all, will indeed be doing so, and lastly

(5) if the game is to be repeated, maintain this confidence

In the tax collection example, (1) means that the taxes are actually collectively beneficial (i.e. better for all of us to be spent by the government than by the free market), (2) means that I have an actual incentive to pay tax as long as I can trust that (at least almost) everyone else is doing the same (by for example increasing the likelihood that I get caught, as the number of cheats goes down). (3) is very very difficult without reforming enforcement from the ground up all at once, (4) means the effectiveness of the measures have to be trusted to work before they even come into force, and (5) means that they actually did work, and can be seen to work, in hindsight. This is why it is so hard for countries, once they have abysmal tax compliance, to attain a state of high tax compliance (almost the only way is by a highly disruptive route, and even then success if far from guaranteed), and yet it is easy to slip into the other direction if the conditions of making the “tax game” a Stag Hunt are not being met. As soon as it turns into a Prisoner’s Dilemma, compliance collapses.

Now we are ready to see the difficulty of enforcing (or even agreeing) global carbon emissions standards.

Condition (1) is fulfilled. We are all better off if we all comply, than if we all don’t comply. However, condition (2) is not fulfilled at all: We lack a credible financial or political penalty for non-compliance, which would be in other countries’ interest to enforce. If hunting the hare is always preferable and cost-less, it is not a Stag Hunt but a Prisoner’s Dilemma: Cooperation cannot be an equilibrium, even if it is strictly efficient, because non-compliance is strictly dominant.

Conditions (3) and (4) fall over (making (5) fall like a domino), because we lack mechanisms for building trust in other countries’ intention to comply. What is worse, compliance costs occur long before you know other countries true intent to comply. The rational strategy is then to pretend willingness to comply while having no true intention to do so. And since we know that his is the optimal strategy for everyone else, there can be no trust that the others will comply eventually.

So on all these scores we see that even nuclear disarmament was a lot easier to negotiate and enforce than global emissions targets, because the one was always a Stag Hunt and the other is a Prisoner’s Dilemma, and because disarmament needed only two countries to build enough trust and monitoring processes, and lastly also because with disarmament treaties compliance was less costly financially than non-compliance. This made the disarmed state a natural, strictly efficient Nash Equilibrium even in the absence of mutual penalties (which, in the international arena, are almost always unlikely to be enforced individually or collectively by the other countries).

Agreements cannot succeed under the current conditions of the emissions targets discussions. We have to find ways to make it a Stag Hunt (making cooperation attractive at least in the state where the cooperation of others can be taken as a matter of trust). This means non-compliance has to have a price that other countries are willing to extract. But let us face it: what, in the real world, is the punishment for non-compliance? And that would only be the first two of five steps. We are missing a big trick here.

[1] except, for the purists amongst the readers, when the game is iterated often amongst the same players

[2] whereas, in a Prisoner’s Dilemma, I am better off defecting even if I know that the other party will cooperate.

[3] This was frequently demonstrated by Hermann Kahn: “Let us assume that we have just been informed that a nuclear device has been dropped in NYC. What do you think the president would do?” Audience: “Press every button for launching nuclear forces and go home”. Kahn: “What happens next?” Audience: “The Soviets do the same” Kahn: “And then?” Audience: “Nothing. Both sides have been destroyed”. Kahn: “Why then would the president do that?”. The threat of mutually assured destruction had a flaw in it: it was not in the interest of either party to carry it out in the event.

One response to “Global Warming is no Stag Hunt

  1. Pingback: Game Theory and the Debt Ceiling | The Sustainability Report·

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