Deer hunting

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The deer hunt is a parable that goes back to Jean-Jacques Rousseau and is also known as the hunting party . In addition, the stag hunt or assurance game , also known as the insurance game , is a fundamental game theory constellation.

Rousseau treated this in the sense of his research on the formation of collective rules under the contradictions of social action, that is, that paradoxical effects lead to the institutionalization of coercion (to cooperation ) so that there is no breach of contract. He describes the situation as follows: Two hunters go on a hunt, each of which has only been able to kill one hare so far. Now they are trying to make arrangements, that is, to make an agreement in order to be able to kill a stag together, which is more profitable for both of them than a single hare.

On the prowl, the dilemma develops analogously to the prisoner's dilemma : If one of the two hunters runs into a hare during the hunt, he has to decide whether to kill the hare or not. If he kills the hare, he misses the opportunity to kill a stag together. At the same time, he has to think about how the other would act. If the other is in the same position, there is a risk that the other will kill the hare and ultimately suffer a loss: neither a hare nor a proportion of a deer.

Deer hunting as a simple symmetrical two-person game with two strategies

Deer hunting is often modeled in game theory as a symmetrical two-person game with two strategies each (deer hunting, hare hunting). The payouts (in utility units) could look like the following payout matrix:

Player 2
Deer hunting Hare hunting
Player 1 Deer hunting 4/4 0/3
Hare hunting 3/0 3/3

(For the purpose of interpreting the matrix, it is assumed that the players will be sure to hunt a stag if they both hunt deer and each one of them will hunt a hare safely if they hunt hare. A single hunter, on the other hand, has no chance to hunt a stag .)

The game has three Nash equilibria . Two in pure strategies (deer hunt / stag hunt and hare hunt / hare hunt) and one in mixed strategies (both players hunt the deer with a probability of 3/4). Without further information about the players, the solution concept of the Nash equilibrium is not sufficient to determine a solution for the game.

However, at least at first glance, the coordination in this game seems simple: the players have agreed on the deer hunt and the result of the deer hunt is better for both players than the result of the two other Nash equilibria, so it is a Pareto efficient Nash equilibrium. (For the comparison with the equilibrium in mixed strategies, the maximization of the expected utility is assumed.)

But there are also arguments against this solution:

  1. If a player is not relatively sure that the other player will actually stay with the deer hunt, it can become rational for him to switch to the hare hunt. More precisely: If he considers the probability that his teammate will hunt rabbits to be greater than 1/4, it would be rational for a hunter who maximizes his expected utility to also hunt rabbits himself.
  2. The hare hunt secures a payout of 3, while the deer hunt carries the risk of a payout of 0. A hunter who wants to be on the safe side could therefore switch to hare hunting. (A hunter who would like to secure a result in this way would not be a maximizer of the expected utility.) Hasenjagd / Hasenjagd is the Maximin solution of the game.

Extension of the model

The simple two-person model with only two strategies leaves out an important element of the motivating story: the change in the situation that occurs when one of the hunters happens to run into a rabbit. In his description of the situation, Rousseau assumes that the hunter prefers the hare's meal, which is now possible earlier, to the venison dish that is only possible later. For him, the (subjective) benefit of hare hunting is higher than the benefit of deer hunting in this case.

In an extended model, therefore, both the trait of nature, which determines which player encounters a rabbit (possibly also both), and the greater benefit of the earlier rabbit meal must be taken into account.

The following matrix shows the situation from the point of view of player 1, who has presumably met a rabbit. The greater benefit of an earlier rabbit meal is rated as +2 (relative to an "ordinary" rabbit meal):

Player 2
no rabbit Met rabbit
Deer hunting Hare hunting Deer hunting Hare hunting
Player 1 Deer hunting 4/4 0/3 4/4 0/5
Hare hunting 5/0 5/3 5/0 5/5

So for player 1, after meeting a hare, it is always better to hunt the hare, regardless of the behavior of the colleague. The strategy of hare hunting is dominant . As a rational player who is only interested in his own well-being, he will therefore certainly chase the hare. Since the situation for player 2 is exactly symmetrical, we know that he, too, will surely chase the rabbit if he hits one.

The next matrix again shows the situation from the point of view of player 1, who, however, presumably has not met a rabbit here :

Player 2
no rabbit Met rabbit
Deer hunting Hare hunting Deer hunting Hare hunting
Player 1 Deer hunting 4/4 0/3 4/4 0/5
Hare hunting 3/0 3/3 3/0 3/5

We already know that player 2 will surely chase the rabbit if he hits one . Player 1's behavior depends on how likely he is that player 2 will hunt a hare . From the analysis of the simple model, we know that player 1 will go hare hunting if he thinks this probability is greater than 1/4. So he is sure to go rabbit hunting if he believes that there is a greater than 1/4 chance that player 2 will hit a rabbit .

As an example, consider a situation in which both hunters know that each of them - regardless of what happens to the other - has a 1/3 chance of hitting a hare. Both hunters will therefore go hare hunting in any case (whether they meet a hare or not). Had they stuck to their agreement, they would have received the (sure) benefit of 4 each. Now everyone gets a benefit of 5 if they happen to meet a rabbit and a benefit of 3 if they also have to find the rabbit first. The expected utility of the two players is 11/3 and is thus lower than the utility of deer hunting. It would be better for both players if they stick to their agreement in any case. As rational players only interested in their own well-being, they will not do that.

Summary

Social action according to rational criteria is often designed in such a way that a certain number of actors, in this case two, have to make a decision for their benefit (or advantage), but separately from one another. I.e. both cannot communicate with each other (during the hunt). The mutual projective reflection of rational decisions may lead to the dilemma of a collectively suboptimal result. If one of the two hunters breaks the contract, they cannot catch the deer. The result of the jointly hunted deer is the optimal result, all other results (for the simple model: both catch a hare or only one catches a hare) are not optimal. For Rousseau, the question arose as to why the actors commit breach of contract against and through their will. This contradiction in social action should therefore give rise to institutions that encourage (force) compliance with the contractual agreements.

literature

  • Jean-Jacques Rousseau: Treatise on the origin and foundations of inequality among people. P. 77 (source in the original text), Reclam, Stuttgart 1998, p. 77, ISBN 978-3150017708
  • Ken Binmore: Game Theory and the Social Contract. Volume 1: Playing Fair. MIT Press, Cambridge MA et al. 1994, ISBN 0-262-02363-6 .
  • Raymond Boudon : Contradictions of Social Action. Luchterhand, Darmstadt et al. 1979, ISBN 3-472-75115-0 ( sociological texts. NF 115).

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