Maximization problem, Game Theory

Two individuals (i ∈ {1, 2}) work independently on a joint project. They each independently decide how much e ort ei they put. E ort choice has to be any real number between 0 and 1 (ei ∈ [0, 1] not just 0 or 1). The cost of putting an amount of e ffort ei is n e2i/2, where n is a parameter greater or equal than 2. If individual i puts e ffort ei, then he succeeds with probability ei and fails with probability 1 - ei. The probability of success of the two agents are independent; this means that both succeed with probability e1x e2, 1 succeeds and 2 fails with probability e1 x(1 - e2), 1 fails and 2 succeeds with probability (1 - e1)e2, and both fail with probability (1 - e1)  (1 - e2).

If at least one of the individuals succeeds then, independently of who did succeed, both individuals get a payo of 1. If none of them succeeds, both individuals get 0. Therefore, each individual is a ected by the action of the other. However, individuals choose the level of e ort that maximizes their own expected utility (bene t minus cost of e ort).

(a) Write down the expected utility of individuals 1 and 2 (note that the utility of 1 depends on the e orts of 1 and 2 and the utility of 2 depends on the e orts of 1 and 2). [Hint. The expected bene t of 1 is the probability that 1 and/or 2 succeed times the payo if 1 and/or 2 succeed plus the probability that both 1 and 2 fail times the payo if both 1 and 2 fail.]

(b) Find the Nash equilibrium of this game, that is, the optimal level of e ort. Find the expected utility of each individual in equilibrium (use the rst-order condition and make sure that the second-order condition is satis ed). Suppose that a benevolent dictator can choose the  level of e ort that both individuals must exert. He chooses the e ort levels that maximize the sum of the expected utilities of both agents (these e orts are also called socially optimal levels).

(c) Write down the maximization problem of the benevolent dictator.

(d) Find the e ort levels that the dictator imposes on each individual (use the rst-order condition and assume that the second-order condition is satis ed). Find the expected utility of each individual.

(e) Compare the e ort level and nal utility of each individual in the cases of Nash Equilibrium (sel sh individual maximization) and benevolent dictatorship.


Posted Date: 3/5/2013 6:38:45 AM | Location : United States

Related Discussions:- Maximization problem, Assignment Help, Ask Question on Maximization problem, Get Answer, Expert's Help, Maximization problem Discussions

Write discussion on Maximization problem
Your posts are moderated
Related Questions
Assuming that there are only 2 airline companies in the world, Delta and US Airways, what is the ((Nash) Equilibrium) or price that each company in the following matrix will charge

I have an assignment in which I have to invent a new international trade theory. For me, the absolute advantage of Adam Smith is really good, and I want to find a solution if a cou

An equilibrium refinement provides how of choosing one or many equilibria from among several in a very game. several games might contain many Nash equilibria, and therefore supply

can i analyse all games under trigger strategies or it''s possible just for prisoners dilemma?

Another term for a preserved bid auction in which bidders simultaneously submit bids to the auctioneer with no knowledge of the amount bid by other member. Usually, the uppermost b

Description The simplest of William Poundstone's social dilemmas during which the every player contains a dominant strategy and also the equilibrium is Pareto optimal. the sole

. A bid is an sign by a potential buyer of the price the buyer is ready to pay for the object being auctioned. In a Procurement Auction, the bid is an sign of the price a seller is

Leadership in an Oil Production Game Students can be broken into pairs to play this game once, witheach student's representing one country; then each shouldswitch partners and

Game Theory has evolved since its start as a thought exercise for academic mathematicians. Taught in economics departments , top business schools, and the strategic analysis, even

A minimum bid is that the smallest acceptable bid in an auction. a gap bid, the primary bid placed within the auction, should be a minimum of as high because the minimum bid or the