Stag hunt
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Stag hunt
In game theory, the stag hunt is a game which describes a conflict between safety and social cooperation. Other
names for it or its variants include "assurance game", "coordination game", and "trust dilemma". Jean-Jacques
Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a
stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual
hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself,
but a hare is worth less than a stag. This is taken to be an important analogy for social cooperation.
The stag hunt differs from the Prisoner's Dilemma in that there are two Nash equilibria: when both players cooperate
and both players defect. In the Prisoners Dilemma, however, despite the fact that both players cooperating is Pareto
efficient, the only Nash equilibrium is when both players choose to defect.
An example of the payoff matrix for the stag hunt is pictured in Figure 2.
Stag
Hare
Stag
Hare
Stag
A, a C, b
Stag
2, 2 0, 1
Hare
B, c
Hare
1, 0 1, 1
D,
d
Fig. 1: Generic stag
hunt
Fig. 2: Stag hunt
example
Formal definition
Formally, a stag hunt is a game with two pure strategy Nash equilibria - one that is risk dominant and another that is
payoff dominant. The payoff matrix in Figure 1 illustrates a stag hunt, where
. Often, games with
a similar structure but without a risk dominant Nash equilibrium are called stag hunts. For instance if a=2, b=1, c=0,
and d=1. While (Hare, Hare) remains a Nash equilibrium, it is no longer risk dominant. Nonetheless many would call
this game a stag hunt.
In addition to the pure strategy Nash equilibria there is one mixed strategy Nash
equilibrium. This equilibrium depends on the payoffs, but the risk dominance condition
places a bound on the mixed strategy Nash equilibrium. No payoffs (that satisfy the
above conditions including risk dominance) can generate a mixed strategy equilibrium
where Stag is played with a probability higher than one half. The best response
correspondences are pictured here.
Stag hunt
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The stag hunt and social cooperation
Although most authors focus on the prisoner's dilemma
as the game that best represents the problem of social
cooperation, some authors believe that the stag hunt
represents an equally (or more) interesting context in
which to study cooperation and its problems (for an
overview see Skyrms 2004).
"Nature and Appearance of Deer" taken from "Livre du Roy Modus,"
created in the 14th century
There is a substantial relationship between the stag hunt
and the prisoner's dilemma. In biology many
circumstances that have been described as prisoner's
dilemma might also be interpreted as a stag hunt,
depending on how fitness is calculated.
Cooperate
Defect
Cooperate
Defect
2, 2
3, 0
0, 3
1, 1
Fig. 3: Prisoner's dilemma
example
It is also the case that some human interactions that seem like prisoner's dilemmas may in fact be stag hunts. For
example, suppose we have a prisoner's dilemma as pictured in Figure 3. The payoff matrix would need adjusting if
players who defect against cooperators might be punished for their defection. For instance, if the expected
punishment is -2, then the imposition of this punishment turns the above prisoner's dilemma into the stag hunt given
at the introduction.
Examples of the stag hunt
The original stag hunt dilemma is as follows: a group of hunters have tracked a large stag, and found it to follow a
certain path. If all the hunters work together, they can kill the stag and all eat. If they are discovered, or do not
cooperate, the stag will flee, and all will go hungry.
The hunters hide and wait along a path. An hour goes by, with no sign of the stag. Two, three, four hours pass, with
no trace. A day passes. The stag may not pass every day, but the hunters are reasonably certain that it will come.
However, a hare is seen by all hunters moving along the path.
If a hunter leaps out and kills the hare, he will eat. However, it results in the trap laid for the stag to be wasted, and
the others will starve. There is no certainty that the stag will arrive; the hare is present. The dilemma is that if one
hunter waits, he risks one of his fellows killing the hare for himself, sacrificing everyone else. This makes the risk
twofold; risk the stag never come, or risk another man taking the kill.
In addition to the example suggested by Rousseau, David Hume provides a series of examples that are stag hunts.
One example addresses two individuals who must row a boat. If both choose to row they can successfully move the
boat. However if one doesn't, the other wastes his effort. Hume's second example involves two neighbors wishing to
drain a meadow. If they both work to drain it they will be successful, but if either fails to do his part the meadow will
not be drained.
Several animal behaviors have been described as stag hunts. One is the coordination of slime molds. In times of
stress, individual unicellular protists will aggregate to form one large body. Here if they all act together they can
successfully reproduce, but success depends on the cooperation of many individual protozoa. Another example is the
hunting practices of orcas (known as carousel feeding). Orcas cooperatively corral large schools of fish to the surface
Stag hunt
and stun them by hitting them with their tails. Since this requires that the fish have no way to escape, it requires the
cooperation of many orcas.
References
• Skyrms, Brian. (2004) The Stag Hunt and Evolution of Social Structure. Cambridge: Cambridge University Press.
External links
• The stag hunt at GameTheory.net [1]
• The stag hunt (pdf) [2] by Brian Skyrms
References
[1] http:/ / www. gametheory. net/ Dictionary/ Games/ StagHunt. html
[2] http:/ / www. lps. uci. edu/ home/ fac-staff/ faculty/ skyrms/ StagHunt. pdf
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Article Sources and Contributors
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Article Sources and Contributors
Stag hunt Source: http://en.wikipedia.org/w/index.php?oldid=416237753 Contributors: AlexP, Applejuicefool, BD2412, Claymore, Clearlyhidden19, CommonsDelinker, Cruci, Derek Ross,
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Image Sources, Licenses and Contributors
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