Stag Hunt
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In game theory, the stag hunt, sometimes referred to as the assurance game, trust dilemma or common interest game, describes a conflict between safety and social cooperation. The stag hunt problem originated with philosopher
Jean-Jacques Rousseau Jean-Jacques Rousseau (, ; 28 June 1712 – 2 July 1778) was a Genevan philosopher, writer, and composer. His political philosophy influenced the progress of the Age of Enlightenment throughout Europe, as well as aspects of the French Revolu ...
in his ''
Discourse on Inequality ''Discourse on the Origin and Basis of Inequality Among Men'' (french: Discours sur l'origine et les fondements de l'inégalité parmi les hommes), also commonly known as the "Second Discourse", is a 1755 work by philosopher Jean-Jacques Roussea ...
''. In Rousseau's telling, two
hunt Hunting is the human practice of seeking, pursuing, capturing, or killing wildlife or feral animals. The most common reasons for humans to hunt are to harvest food (i.e. meat) and useful animal products (fur/ hide, bone/tusks, horn/antler, e ...
ers must decide separately, and without the other knowing, whether to hunt a stag or a hare. However, both hunters know the only way to successfully hunt a stag is with the other's help. One hunter can catch a hare alone with less effort and less time, but it is worth far less than a stag and has much less meat. Rousseau therefore posits it would be much better for each hunter, acting individually, to give up total autonomy and minimal risk, which brings only the small reward of the hare. Instead, each hunter should separately choose the more ambitious and far more rewarding goal of getting the stag, thereby giving up some autonomy in exchange for the other hunter's cooperation and added might. Commentators have seen the situation as a useful analogy for many kinds of social cooperation, such as international agreements on climate change. The stag hunt differs from the
prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("def ...
in that there are two pure-strategy
Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
: one where both players cooperate, and one where both players defect. In the Prisoner's Dilemma, in contrast, despite the fact that both players cooperating is
Pareto efficient Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
, the only pure Nash equilibrium is when both players choose to defect. An example of the
payoff matrix In game theory, normal form is a description of a ''game''. Unlike extensive form, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyin ...
for the stag hunt is pictured in Figure 2.


Formal definition

Formally, a stag hunt is a game with two
pure strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
Nash equilibria—one that is
risk dominant Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superio ...
and another that is payoff dominant. The
payoff matrix In game theory, normal form is a description of a ''game''. Unlike extensive form, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyin ...
in Figure 1 illustrates a generic stag hunt, where a>b\ge d>c. Often, games with a similar structure but without a risk dominant Nash equilibrium are called assurance games. For instance if ''a''=10, ''b''=5, ''c''=0, and ''d''=2. 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 In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
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 In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given (; ). The concept of a best response is central to John Nash's best-known contribu ...
correspondences are pictured here.


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 ). There is a substantial relationship between the stag hunt and the prisoner's dilemma. In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
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.


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, but the trap laid for the stag will be wasted and the other hunters 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; the risk that the stag does not appear, and the risk that another hunter takes the kill. In addition to the example suggested by Rousseau,
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment phil ...
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 mold Slime mold or slime mould is an informal name given to several kinds of unrelated eukaryotic organisms with a life cycle that includes a free-living single-celled stage and the formation of spores. Spores are often produced in macroscopic mul ...
s. 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 The orca or killer whale (''Orcinus orca'') is a toothed whale belonging to the oceanic dolphin family, of which it is the largest member. It is the only extant species in the genus '' Orcinus'' and is recognizable by its black-and-white ...
s (known as
carousel feeding Carousel feeding is a cooperative hunting method used by Norwegian orcas ( ''Orcinus orca'') to capture wintering Norwegian spring-spawning herring (''Clupea harengus)''. The term carousel feeding was first used to describe a similar hunting behavi ...
). 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. Author
James Cambias James Cambias (also as James L. Cambias) is an American science fiction and fantasy writer and tabletop game designer. Biography Born and raised in New Orleans, Louisiana, Cambias became interested in space and astronomy at a young age. He ...
describes a solution to the game as the basis for an extraterrestrial civilization in his 2014 science fiction book ''A Darkling Sea''.
Carol M. Rose Carol M. Rose is the Ashby Lohse Chair in Water and Natural Resources at the University of Arizona James E. Rogers College of Law and was previously the Gordon Bradford Tweedy Professor of Law and Organization at Yale Law School. Rose graduated fro ...
argues that the stag hunt theory is useful in 'law and humanities' theory. In international law, countries are the participants in a stag hunt. They can, for example, work together to improve good corporate governance.


See also

*
Common knowledge (logic) Common knowledge is a special kind of knowledge for a group of agents. There is ''common knowledge'' of ''p'' in a group of agents ''G'' when all the agents in ''G'' know ''p'', they all know that they know ''p'', they all know that they all know ...
* ''
Discourse on Inequality ''Discourse on the Origin and Basis of Inequality Among Men'' (french: Discours sur l'origine et les fondements de l'inégalité parmi les hommes), also commonly known as the "Second Discourse", is a 1755 work by philosopher Jean-Jacques Roussea ...
'' *
Mutual knowledge Mutual knowledge in game theory is information known by all participatory agents. However, unlike common knowledge, a related topic, mutual knowledge does not require that all agents are aware that this knowledge is mutual. All common knowledge is ...
*
Pluralistic ignorance In social psychology, pluralistic ignorance refers to a situation in which the minority position on a given topic is wrongly perceived to be the majority position or where the majority position is wrongly perceived to be the minority position. Thi ...
*
Prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("def ...
* Social contract


References

;Notes ;Bibliography *


External links


The stag hunt at GameTheory.net
* {{DEFAULTSORT:Stag Hunt Non-cooperative games Evolutionary game theory Social science experiments