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
Rollback shows that Boeing chooses peace over war if Airbus enters, so Airbus will enter. Rollback equilibrium entails Airbus playing “Enter” and Boeing playing “Peace if entry”; e

A bidding increment is defined by the auctioneer as the least amount above the previous bid that a new bid must be in order to be adequate to the auctioneer. For example, if the in

#questi1 A, Explain how a person can be free to choose but his or her choices are casually determined by past event 2 B , Draw the casual tree for newcomb''s problem when Eve ca

how do tron legacy made?

What do you study about the saving, investment spending and financial system? Savings, Investment Spending, and the Financial System: 1. The correlation between savings and

a) Show that A counting proof could be fun(?). But any old proof will do. (Note that the coefficients (1,2,1) in the above are just the elements of the second row of Pas

Assurance game Scenario "Assurance game" may be a generic name for the sport a lot of commonly called "Stag Hunt." The French thinker, Jean Jacques Rousseau, presented the subse

An auction during which many (more than one) things are offered for sale. Mechanisms for allocating multiple units embody discriminatory and uniform worth auctions.

Scenario Two hooligans with one thing to prove drive at one another on a slender road. the primary to swerve loses faces among his peers. If neither swerves, however, a terminal

Case study GAME 1 Rock-Scissors-Paper This game entails playing three different versions of the children's game rock-scissors-paper. In rock-scissors-paper, two people si