Assume a town that stretches out along a main road that is 1 mile long. There ar

Question

Assume a town that stretches out along a main road that is 1 mile long. There are two companies (A & B) that have decided to open up gas stations along this mile stretch.

Assume that customers are uniformly distributed along the mile and that they have no preference for one brand of gasoline over the other. Customers will simply purchase from the gas station that is closest to them. Where should each company set their gas station if they are to attract the most customers?

(Picture a straight line, with a line in the middle indicating 0.5 -- Now, where should A & B gas stations be located within that mile?)

Assume next that the two gas stations compete by setting their price. Since there is no notable difference between the two brands of gasoline, customers will purchase gas from the station with the cheapest prices. Each gas station has two possible strategies: High Price or Low Price. The market value at a high price is $900,000. The market value at a low price is $500,000. When both stations charge the same price, each station gets 50% of the customers and consequently 50% of the market value. Construct a matrix for the game between both gas stations. Make sure to identify the players, the strategies and the payoffs.

Find the Nash Equilibrium of the game and explain why your result is the equilibrium. If the Nash Equilibrium the best outcome for the game? If not, explain how this outcome can be improved.

Explanation / Answer

A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy. In game theory, the Nash equilibrium is a solution concept of a non-cooperative 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.[1]:14 If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. Stated simply, Amy and Wili are in Nash equilibrium if Amy is making the best decision she can, taking into account Wili's decision, and Wili is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others.Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. In other words, it provides a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do, taking into account the decision-making of the others. Nash equilibrium has been used to analyze hostile situations like war and arms races[2] (see prisoner's dilemma), and also how conflict may be mitigated by repeated interaction (see tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). It has been used to study the adoption of technical standards[citation needed], and also the occurrence of bank runs and currency crises (see coordination game). Other applications include traffic flow (see Wardrop's principle), how to organize auctions (see auction theory), the outcome of efforts exerted by multiple parties in the education process,[3] regulatory legislation such as environmental regulations (see tragedy of the Commons),[4] and even penalty kicks in soccer (see matching pennies)

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