Game Theory Nash Equilibrium
Issues in Microeconomics
Game Theory Coursework 1
By Paul Allen
What is the Nash Equilibrium? This is the idea that equilibrium in a physical system, was that players would adjust their strategies until no player could benefit from changing. All players are then choosing strategies that are best (utility maximising) responses to all other players strategies.(i)
This concept is a set of strategies where any given player can’t get a better payoff by changing their strategy if all players keep their strategic choices constant. Thus once our hypothetical market reaches a Nash Equilibrium, it will be stable, since no individual strategic response would change anything.
One reason to choose the most desirable multiple Nash equilibria on the basis of perfect rationality is the Pareto optimality concept. A Pareto optimum is an equilibrium for which no player can get better off without making the other player worse off.
In game theory we assume rational behaviour by all players; players are perfectly informed and follow their best strategy. A consistent ranking will be given to all the possible payoffs from chosen strategies. The rationality can be said in two different counts.
- Complete knowledge of self-interests exists.
- Meticulously calculated actions of what interests serve best are considered.
This of course is not the case there are systematic errors. An idiot behaving in hast, or an insane person repeating the same process expecting a different result.
- A rational player will move more conservatively.
- A maximum strategy to expose lose may be selected.
Nash equilibrium is rarely Pareto efficient why? Dominant strategy equilibria are when there is one optimal choice of strategy for both players no matter what the other player does this rarely happens. Often there is no not one Nash Equilibrium but numerous equilibriums.
This occurs when one player’s best response to any strategy the other player may pick results in the highest playoff for the players. The players best strategy is not dependant on another player this makes it a dominant strategy. Nash equilibrium exists when one player has a dominant strategy, letting all other players respond with their best alternative.
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- Strongly dominant strategies yields a greater payoff from all other strategies, no matter what the other opponents chooses.
- Weekly dominant strategy is at least as good as all other strategies no matter what other opponents choose.
The alternatives of strategies are randomly mixed. The players mix various strategies to maximise payoff. It is the probability distribution over or some of all the strategies available to players.
If we consider the game show split or steal we have the perfect example of game theory. The example above shows this scenario is not dominance solvable. Neither strategy is dominant for either player because there is no one strategy that is always best. If this game was repeated it would be in the two player’s best interest to cooperate.
- Pareto Equilibrium Player A shares , Player B shares
- Nash Equilibrium Player A steals, Player B shares/Player B steals and Player A shares.
This here game is a one of game, yet there is incentive to cooperate. It’s the classic prisoner’s dilemma. If one of the two players stops cooperating there is 50% possibility a higher payoff. Then it becomes a zero sum game, the person gains but the other person must lose.
What is Prisoners Dilemma?
To expand on the small piece of information offered on prisoner’s dilemma it’s a simple concept of applied mathematics of game theory. We assume rationality of the player, it can be assumed each player seeks to maximise their own payoff without thinking about the cost to society. This means both parties can be left worse off than they could have been.
This video happens to be the perfect example of prisoners’ dilemma and one of my favourite shows. The two prisoners have an incentive to collude with each other and say nothing. If this happens the two would receive smaller sentences. There is a huge incentive for both to betray each other. The Nash equilibrium is if one player cheats on the other, hoping the other stays loyal. The Pareto optimal choice is for both to remain loyal to each other. If both seek to follow their own self-interest as the pair serve five years each. The prisoners dilemma game’s equilibrium gives a sub-optimal Pareto solution meaning that rational leads the players to betray one another yet if they collude with each other it would be better for each player.
Why the prisoner’s dilemma is avoided if the game is repeated infinitely?
The prisoner’s dilemma is avoided when repeated infinitely as both players will come to the realization that it’s better to collude than to cheat one another.
Tit for Tat
Whilst we assume both parties are rational they will seek to enhance their own utility. If they think only of their own selfish needs then you can assume that neither side will cooperate again. This is known as the grim trigger or death spiral.
(ii) “In the single shot context, the prospects for cooperation arte dismal, yet in repeated play context, cooperation is possible if nations condition future play on past behaviour. Grim trigger is a common example of such a strategy. Under the grim trigger, nations initially cooperate and continue to do so provided that defection is never observed. If either player ever defects, then nations refuse to cooperate in every subsequent period. Since the threat of punishment, the indefinite removal of cooperation, is the harshest possible penalty for non-cooperation, the grim trigger strategy represents the limiting case.”
If we use the example of the two prisoners again, one prisoner betrays the other. Let’s assume both prisoners are again in the same situation. There is now a distrust and the betrayed prisoner will no longer wish to collude but will betray the other prisoner. This here behaviour is known as tit for tat.
Let’s now assume that this is a never ending cycle for the two prisoners. Well they no betraying one another will reduce their payoff. Both parties knowing that it is a never ending cycle will now look to collude. This is known as reciprocal altruism as one good turn will therefore be passed on to them eventually. The best thing to do is to help one another, if not than both suffer and since it’s a repeated game than the belligerent party will then seek retaliation. You would use backward induction you look at all possible scenarios you look ahead to the future and work backwards from all possible outcomes and pick the most favourable one. (iii)
In the 1960s, tobacco producers engaged in fierce battles for market share. The major weapon in that war was advertising — advertising that was designed not to attract new smokers, but to lure smokers away from competing brands. Consider the following scenario: There are two tobacco sellers, Phillip and R. J., each of whom can choose to advertise on TV (at a cost of $20 million) or not. There are $100 million of pre-advertising profits available to the two firms. If they both adopt the same budget, they will split the market evenly. If one chooses a high budget while the other chooses low, the high-budget firm will steal half the other’s customers and capture $75 million of pre-advertising profit; the other will earn $25 million. The firms’ net profits (after advertising expenses are considered) are illustrated in the payoff matrix below:
Althea backward induction table verifies the story told above the diagram is in fact true. There are $100 million available profits. If both choose not to advertise than profits are split 50/50. If both choose to advertise they spend $20 million and gain $25 million leaving the advertising company with $55 and belligerent company with $25million profit. Both advertise spending $20 million each, leaving them with $60million profit available which is split 50/50 so both companies earn $30 each.
B. There are two Nash Equilibriums, when one company decided to advertise and the other doesn’t leaving the profits $55, $25. There is first mover disadvantage as they can’t find themselves in a Nash Equilibrium. Both firms are rational so if the first firm tries to collude and don’t advertise the second firm will seek to maximise profits and betray the first firm. If the first firm decides to advertise well the other firm must advertise meaning neither firm will be in a Nash Equilibrium. So only the second mover firm can benefit from being in a Nash Equilibrium in the Short Run.
Cliff both firms promise not to advertise this can be a credible claim. The two firms are rivals and will continue to be rivals for years to come. They both seek to maximise profits so using backwards induction they can see both firms not advertising maximises profits. If this game was a one of game than rational would force the firms to betray each other. As they are both rational firms than they understand this would lead to a death spiral. The two firms would copy each other’s last move leading to distrust this is also known as the grim trigger.
D. The firm’s complaints are not genuine because it benefits them better to not advertise as they both gain $20 more in profit because they don't have any advertising costs. It also means that tit for tat strategy is no longer an issue for the firms. If one firm is in a Nash Equilibrium than it means the other will next time advertise tit for tat strategy which isn’t beneficial in the long run.
- Behavioural Game Theory written by Colin F.Camerer 2003.
Text used from chapter one introduction page two.
- McGillivray, Fiona, and Alastair Smith. "Trust and cooperation through agent-specific punishments." International Organization 54.4 (2000): 809-824.
- Benoit, Jean-Pierre, and Vijay Krishna. "Finitely repeated games." Econometrica: Journal of the Econometric Society (1985): 905-922.
The class notes of George Tridimas were also used in this essay.