Stag Hunt Essay Final - Grade: 73 PDF

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Summary

Choose ONE of the following games:
(a) The Prisoners’ Dilemma (b) Stag Hunt (c) Hawk-Dove
For the game you chose, assess its significance for the social sciences and comment on its theoretical equilibrium (or equilibria)....

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Cho os eONEo ft hef o l l o wi ngg ame s : ( a )ThePr i s o ne r s ’Di l e mma( b)St agHunt( c )Ha wk Do v e Fo rt heg a mey o uc hos e , a s s e s si t ss i g ni fic anc ef ort hes o c i al s c i e nc e sa ndc o mme nto ni t st he o r e t i c a le qui l i br i um ( o r e qui l i br i a ) .

Student ID: 16051802 Module: Game Theory and Economic Applications Module Leader: Tassos Patokos Module Code: 6BUS1140-0901-2019 Word Count: 1, 64 9wo r d s

The origin of Stag Hunt begun with the philosopher Jean-Jacques Rousseau who used it as a basis to describe social cooperation. This consisted of two hunters, each with the choice of hunting a rabbit or a stag. The stag is the optimal prize as it is larger and the rabbit is perceived as the less optimal choice, however, it is still satisfying. The stag can only be caught if both individuals choose the strategy stag, thus this combination results both players being rewarded. In the static game, if one of the players attempts to catch the stag and the other opts for the rabbit then the resultant utility is nothing for the former. However, this is not detrimental to the latter as they have no issues catching the rabbit and still receive a payoff. The final combination is one where both players hunt the rabbit and experience equal payoffs, resulting in the payoff matrix in the below. According to Easley and Kleinberg (2010), the Stag Hunt game captures some of the intuitive challenges raised by the Prisoner’s Dilemma. The structures are clearly different, since the Prisoner’s Dilemma has strictly dominant strategies; both, however, have the property that both players can benefit through cooperation, but risk suffering if they try cooperating while their partner doesn’t (Easley and

(Krueger, 2016)

Using Stag Hunt, the element of social interaction is emphasised as both players find it optimal to cooperate only if the other player cooperates as well. Likewise, if one hunter does not want to cooperate then it is best for the other hunter to also take this strategy.

From this analysis it is evident that there are three Nash equilibria. Two Nash equilibria in pure strategies; when both players choose stag and the other when both players choose rabbit. There is also a Nash equilibrium in mixed strategies when each hunter chooses stag with probability 0.5 and rabbit with probability 0.5. This means both players are indifferent between cooperation and defection and was used by Rousseau as a means of emphasising the power of optimism within a group. The existence of multiple Nash equilibria leads to the problem of indeterminacy. The situation where both players choose to cooperate is mutually

beneficial and the pareto optimal choice, yet, it is not a unique equilibrium to the uncertainty of each player. Ultimately, what each player believes the other will do determines their personal degree of optimism. In an ideal world both players would enjoy the outcome of hunting the stag, nonetheless, this is not achievable with uncertainty. Thus, if one player is doubtful then their best option is to not cooperate, resulting in a sub-optimal solution for both. The stag strategy is clearly one that takes president, therefore making it a likely focal point and overall the strategy that should be chosen. In the case of experimental case, according to Goeree and Holt (2001) ‘A Matching Pennies Game’ can be attributed to comparing the theory with reality. The game consisted of a row player with the ability to choose between top and bottom and the column player with the option of left or right. The symmetric game first played had choice percentages of fifty-fifty meaning the players are indifferent between the two alternatives. In order to make it an asymmetric game the row players payoff was increased and since the column player payoffs were unchanged, the mixed-strategy Nash equilibrium predicts that row’s decision probabilities should remain unchanged as well. However, the data significantly rejects this with 96% of row players choosing the new high payoff (Goeree and Holt, 2001). The interesting factor in this is the column players anticipated this due to them playing right 84% of the time. Highlighting the fact that a variation in payoffs in reality leads to players responding to their own utility accordingly and that Nash mixed strategy prediction is only effective when payoffs are symmetric. As a whole, this leads to the synopsis that the theory is not consistent in reality due to the human factor leading to contradictory results. A real-life example of the Stag Hunt is highlighted by Skyrms (2001) who stated David Hume also has the Stag Hunt. His most famous illustration of a convention has the structure of a two-person Stag Hunt game: two men who pull at the oars of a boat, do it by an agreement or convention, tho' they have never given promises to each other... Both men can either row or not row. If both row, they get the outcome that is best for each – just as in Rousseau’s example (Skyrms, 2001). Here, both men have the choice between receiving the highest payoff of the boat moving or not, similar to the choice in Stag Hunt so it can be used to model real life interactions. In addition, another example that can be attributed is the practices of climate control. The idea behind this is that two countrie have the option to pollute or to not, in other words, both can agree to reduce their individual pollution and reap the rewards. Relating this to Stag Hunt

means the two countries can either cooperate or defect and if one country chose to not cooperate and simply not care about the effects of climate change then there would be no incentive for the other country to reduce their pollution. Therefore, leading to a sup-optimal result. The dynamic version of the Prisoners Dilemma can be resolved through backward induction, however, the same cannot be said for Stag Hunt relying on solely on this. The prominent difference is that, in Stag Hunt, there will be mutual cooperation and mutual defection between players. But, backwards induction can still be applied. In the dynamic version of this game the two players randomise between two strategies with equal probabilities below.

(Patokos, 2019)

According to Heap and Varoufakis (2004) a ‘rational solution’ corresponds to a set of strategies that the players of the game have no reason to regret choosing. Player one believes player two is rational and so compares a payoff of three with one. This means player one would prefer to cooperate at t=1. Likewise, player one also knows that player two will defect if they defect in the first instance. Due to the fact player one cooperates at t=1 so will player two, leading to both players receiving a payoff of three each and therefore the subgame perfect Nash equilibrium is cooperation by both players.

From the conclusion made on the dynamic version of Stag Hunt, the vital ingredient was that each player must use a combination of backward induction with the assumption of common knowledge of rationality. Through backwards induction it was clear to see what player two would want to do and so when the first period was reached player one could do the same. Without this the subgame perfect Nash equilibrium could not be achieved. One significant point to mention is that the subgame perfect Nash equilibrium of Stag Hunt is unique, meaning the analysis of the dynamic version enables a refinement of the game. It allows the problem of indeterminacy to be removed due to the presence of more than one Nash equilibria, resulting in a state of fewer Nash equilibria. Therefore, it can be argued the

dynamic version of Stag Hunt provides a more accurate analysis of the optimal choice for players in comparison to the static game.

The argument that mutual cooperation is an attraction to both players and thus a Schelling point is clear, yet, individuals always remain hesitant. The idea of selfish incentives means both players wanting the stag can produce social cooperation. Nevertheless, the stag can be seen as risky as hunting the rabbit will always provide a payoff regardless, meaning the perspective of a social planner must be considered. The sole purpose of a social planner would be to create an environment that fosters cooperation through setting a law stating that ‘both players must cooperate at all times.’ As we already know, mutual cooperation occurs in the theory, however, the experimental evidence has shown that the human factor can change this. Therefore, the setting of this law ensures that regardless of any payoff cooperation will always be achieved. Another solution could be the use of rewards to incentivise individuals into cooperation. If players know they will receive extra food for hunting the stag, then they are more likely to cooperate meaning an effective solution can be attained. Subsequently, the idea of laws through social planning fundamentally links in with Rousseau’s idea of a social contract. He argued that a contract should be formed that binds people into a community that exists for mutual preservation.

Finally, the study of repeated games where the end period is unknown enables the analysis of the repeated version of Stag Hunt to be made. This situation contains the problem of indeterminacy, similar to static version of Stag Hunt which consists of multiple equilibria in pure strategies. The Folk theorem suggests that any strategy, even strictly dominated ones, can be considered a Nash equilibrium. Due to this theorem it makes it almost ineffective to study the repeated version of Stag Hunt and is the reason many mainstream economists try to not invest too much time into linking their applications with this theorem. To conclude, the three Nash equilibria in the static version of the Stag Hunt game have been clearly identified with two in pure strategies and one in mixed. Along with this is the combination of backwards induction and common knowledge of rationality in order to reach the subgame perfect Nash equilibrium of mutual cooperation. The idea of mutual cooperation was used to highlight it as a Schelling point, however, the problem of uncertainty plays a key role. As a result, the need for social planning was emphasised with the idea of laws and incentives put in place in order to ensure players select the optimal strategy.

References:

Easley, D. and Kleinberg, J. (2010). Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, pp.155-171.

Goeree, J. and Holt, C. (2001). Ten Little Treasures of Game Theory and Ten Intuitive Contradictions. American Economic Review, 91(5), pp.1402-1422.

Heap, S. and Varoufakis, Y. (2004). Game theory, second edition. 2nd ed. London: Routledge, p.41.

Krueger,

J.

(2016). Duero

Dilemma.

[image]

Available

at:

https://www.psychologytoday.com/za/blog/one-among-many/201612/game-lunch-and-love [Accessed 9 Dec. 2019].

Patokos, T. (2019). Week 6: Dynamic games, subgame perfection.

Skyrms, B. (2001). The Stag Hunt. Proceedings and Addresses of the American Philosophical Association, 75(2), 31-41....