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36-520: [REDACTED] Look up desirable in Wiktionary, the free dictionary. Desirable may refer to: something that is considered a favorable outcome, see e.g. best response Desirable (film) Desirable (horse) , a racehorse See also [ edit ] Desire (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

72-595: 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

108-413: A conflict between safety and social cooperation. The stag hunt problem originated with philosopher Jean-Jacques Rousseau in his Discourse on Inequality . In the most common account of this dilemma, which is quite different from Rousseau's, two hunters 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

144-405: 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

180-488: 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 generic stag hunt, where a > b ≥ d > c {\displaystyle a>b\geq d>c} . In addition to the pure strategy Nash equilibria there is one mixed strategy Nash equilibrium. This equilibrium depends on

216-403: 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

252-532: 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. But both hunters would be better off if both choose the more ambitious and more rewarding goal of getting the stag, giving up some autonomy in exchange for the other hunter's cooperation and added might. This situation is often seen as a useful analogy for many kinds of social cooperation, such as international agreements on climate change. The stag hunt differs from

288-564: Is illustrated in Figure 8, where black represents the best response correspondence and the other colors each represent different smoothed best response functions. In standard best response correspondences, even the slightest benefit to one action will result in the individual playing that action with probability 1. In smoothed best response as the difference between two actions decreases the individual's play approaches 50:50. There are many functions that represent smoothed best response functions. The functions illustrated here are several variations on

324-474: 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 provides a series of examples that are stag hunts. One example addresses two individuals who must row

360-426: Is not really a game theoretical problem). Any payoff symmetric 2 × 2 game will take one of these three forms. Games in which players score highest when both players choose the same strategy, such as the stag hunt and battle of the sexes , are called coordination games . These games have reaction correspondences of the same shape as Figure 3, where there is one Nash equilibrium in the bottom left corner, another in

396-407: Is said to exist, and the corner Nash equilibria are ESSes . Games with dominated strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric 2 × 2 games. For instance, in the single-play prisoner's dilemma , the "Cooperate" move is not optimal for any probability of opponent Cooperation. Figure 5 shows

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432-416: Is the matching pennies game. In this game one player, the row player (graphed on the y dimension) wins if the players coordinate (both choose heads or both choose tails) while the other player, the column player (shown in the x -axis) wins if the players discoordinate. Player Y's reaction correspondence is that of a coordination game, while that of player X is a discoordination game. The only Nash equilibrium

468-416: Is the combination of mixed strategies where both players independently choose heads and tails with probability 0.5 each. In evolutionary game theory , best response dynamics represents a class of strategy updating rules, where players strategies in the next round are determined by their best responses to some subset of the population. Some examples include: Importantly, in these models players only choose

504-452: Is −2, then the imposition of this punishment turns the above prisoner's dilemma into the stag hunt given at the introduction. 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

540-420: The optimal probability that player Y plays 'Stag' (in the y -axis), as a function of the probability that player X plays Stag (shown in the x -axis). In Figure 2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x -axis), as a function of the probability that player Y plays Stag (shown in the y -axis). Note that Figure 2 plots the independent and response variables in

576-408: The prisoner's dilemma in that there are two pure-strategy Nash equilibria : one where both players cooperate, and one where both players defect. In the prisoner's dilemma, despite the fact that both players cooperating is Pareto efficient , the only pure Nash equilibrium is when both players choose to defect. An example of the payoff matrix for the stag hunt is pictured in Figure 2. Formally,

612-435: The basis for an extraterrestrial civilization in his 2014 science fiction book A Darkling Sea . Carol M. Rose 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. Robert Aumann proposed: "Let us now change the scenario by permitting pre-play communication. On

648-414: The best response for every player: Theorem  —  In any finite potential game, best response dynamics always converge to a Nash equilibrium. Instead of best response correspondences, some models use smoothed best response functions . These functions are similar to the best response correspondence, except that the function does not "jump" from one pure strategy to another. The difference

684-428: The best response on the next round that would give them the highest payoff on the next round . Players do not consider the effect that choosing a strategy on the next round would have on future play in the game. This constraint results in the dynamical rule often being called myopic best response . In the theory of potential games , best response dynamics refers to a way of finding a Nash equilibrium by computing

720-400: 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 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 describes a solution to the game as

756-410: The degree to which the function deviates from the true best response (a larger γ implies that the player is more likely to make 'mistakes'). There are several advantages to using smoothed best response, both theoretical and empirical. First, it is consistent with psychological experiments; when individuals are roughly indifferent between two actions they appear to choose more or less at random. Second,

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792-414: The face of it, it seems that the players can then 'agree' to play (c,c); though the agreement is not enforceable, it removes each player's doubt about the other one playing c". Aumann concluded that in this game "agreement has no effect, one way or the other." It is his argument: "The information that such an agreement conveys is not that the players will keep it (since it is not binding), but that each wants

828-412: The following function: e E ( 1 ) / γ e E ( 1 ) / γ + e E ( 2 ) / γ {\displaystyle {\frac {e^{E(1)/\gamma }}{e^{E(1)/\gamma }+e^{E(2)/\gamma }}}} where E ( x ) represents the expected payoff of action x , and γ is a parameter that determines

864-525: The opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the Nash equilibria at the points where the two player's best responses agree in Figure 3. There are three distinctive reaction correspondence shapes, one for each of the three types of symmetric 2 × 2 games: coordination games, discoordination games, and games with dominated strategies (the trivial fourth case in which payoffs are always equal for both moves

900-565: The other player plays 2 is above threshold), and indifferent (both strategies play equally well under all conditions). While there are only four possible types of payoff symmetric 2 × 2 games (of which one is trivial), the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other. The dimensions may be redefined (exchange names of strategies 1 and 2) to produce symmetrical games which are logically identical. One well-known game with payoff asymmetries

936-427: The other to keep it." In this game "each player always prefers the other to play c, no matter what he himself plays. Therefore, an agreement to play (c,c) conveys no information about what the players will do, and cannot be considered self-enforcing." Weiss and Agassi wrote about this argument: "This we deem somewhat incorrect since it is an oversight of the agreement that may change the mutual expectations of players that

972-419: 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. Although most authors focus on the prisoner's dilemma as the game that best represents

1008-497: The play of individuals is uniquely determined in all cases, since it is a correspondence that is also a function . Finally, using smoothed best response with some learning rules (as in Fictitious play ) can result in players learning to play mixed strategy Nash equilibria . Stag hunt In game theory , the stag hunt , sometimes referred to as the assurance game , trust dilemma or common interest game , describes

1044-422: The player's strategies. So, for any given set of opponent's strategies σ −i , b i ( σ −i ) represents player i ' s best responses to σ −i . Response correspondences for all 2 × 2 normal form games can be drawn with a line for each player in a unit square strategy space . Figures 1 to 3 graphs the best response correspondences for the stag hunt game. The dotted line in Figure 1 shows

1080-408: 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

1116-404: The proof of the existence of mixed strategy Nash equilibria. Reaction correspondences are not "reaction functions" since functions must only have one value per argument, and many reaction correspondences will be undefined, i.e., a vertical line, for some opponent strategy choice. One constructs a correspondence b (·) , for each player from the set of opponent strategy profiles into the set of

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1152-466: The reaction correspondence for such a game, where the dimensions are "Probability play Cooperate", the Nash equilibrium is in the lower left corner where neither player plays Cooperate. If the dimensions were defined as "Probability play Defect", then both players best response curves would be 1 for all opponent strategy probabilities and the reaction correspondences would cross (and form a Nash equilibrium) at

1188-537: The title Desirable . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Desirable&oldid=1092161137 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Best response Reaction correspondences , also known as best response correspondences, are used in

1224-476: The top left and bottom right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a mixed strategy which lies along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixed Nash is an evolutionarily stable strategy (ESS) , as play is confined to the bottom left to top right diagonal line. Otherwise an uncorrelated asymmetry

1260-444: The top right corner. A wider range of reaction correspondences shapes is possible in 2 × 2 games with payoff asymmetries. For each player there are five possible best response shapes, shown in Figure 6. From left to right these are: dominated strategy (always play 2), dominated strategy (always play 1), rising (play strategy 2 if probability that the other player plays 2 is above threshold), falling (play strategy 1 if probability that

1296-418: The top right, and a mixing Nash somewhere along the diagonal between the other two. Games such as the game of chicken and hawk-dove game in which players score highest when they choose opposite strategies, i.e., discoordinate, are called anti-coordination games. They have reaction correspondences (Figure 4) that cross in the opposite direction to coordination games, with three Nash equilibria, one in each of

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