Stag hunt
Encyclopedia
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 Prisoner's Dilemma, in contrast, 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.
- 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.
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).
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.
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.
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 coming, 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 orca
s (known as carousel feeding
). Orcas cooperatively corral large schools of fish to the surface 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.
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, 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
Jean-Jacques Rousseau
Jean-Jacques Rousseau was a Genevan philosopher, writer, and composer of 18th-century Romanticism. His political philosophy influenced the French Revolution as well as the overall development of modern political, sociological and educational thought.His novel Émile: or, On Education is a treatise...
described a situation in which two individuals go out on a hunt
Hunting
Hunting is the practice of pursuing any living thing, usually wildlife, for food, recreation, or trade. In present-day use, the term refers to lawful hunting, as distinguished from poaching, which is the killing, trapping or capture of the hunted species contrary to applicable law...
. Each can individually choose to hunt a stag
STAG
STAG: A Test of Love is a reality TV show hosted by Tommy Habeeb. Each episode profiles an engaged couple a week or two before their wedding. The cameras then follow the groom on his bachelor party...
or hunt a hare
Hare
Hares and jackrabbits are leporids belonging to the genus Lepus. Hares less than one year old are called leverets. Four species commonly known as types of hare are classified outside of Lepus: the hispid hare , and three species known as red rock hares .Hares are very fast-moving...
. 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
Prisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
in that there are two Nash equilibria: when both players cooperate and both players defect. In the Prisoner's Dilemma, in contrast, despite the fact that both players cooperating is Pareto efficient, the only Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
is when both players choose to defect.
An example of the payoff matrix for the stag hunt is pictured in Figure 2.
Formal definition
Formally, a stag hunt is a game with two pure strategy Nash equilibriaNash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
- 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 equilibriumNash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
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.
Best response
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given...
correspondences are pictured here.
The stag hunt and social cooperation
Although most authors focus on the prisoner's dilemmaPrisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
as the game that best represents the problem of social cooperation
Cooperation
Cooperation or co-operation is the process of working or acting together. In its simplest form it involves things working in harmony, side by side, while in its more complicated forms, it can involve something as complex as the inner workings of a human being or even the social patterns of a...
, 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).
There is a substantial relationship between the stag hunt and the prisoner's dilemma. In biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
many circumstances that have been described as prisoner's dilemma might also be interpreted as a stag hunt, depending on how fitness is calculated.
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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
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 coming, or risk another man taking the kill.
In addition to the example suggested by Rousseau, David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
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 orca
Orca
The killer whale , commonly referred to as the orca, and less commonly as the blackfish, is a toothed whale belonging to the oceanic dolphin family. Killer whales are found in all oceans, from the frigid Arctic and Antarctic regions to tropical seas...
s (known as carousel feeding
Carousel feeding
Carousel feeding refers to the hunting technique commonly employed by orcas. An individual or small group of orcas will release bursts of bubble or flash their white undersides to round prey such as salmon or herring into a tight defensive ball close to the surface. The orcas then slap the ball...
). Orcas cooperatively corral large schools of fish to the surface 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.
External links
- The stag hunt at GameTheory.net
- The stag hunt (pdf) by Brian Skyrms