Question

EXERCISE 149.2 (Normalized Bernoulli payoff functions) Suppose that a decision- maker's preferences can be represented by the expected value of the Bernoulli pay- off function u. Find a Bernoulli payoff function whose expected value represents the decision-maker's preferences and assigns a payoff of 1 to the best outcome and a payoff of 0 to the worst outcome.

please answer step by step

Answer 1

A payoff function is a function:

F: S₁*S₂*...Sm$\rightarrow$R

whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set P= {1, 2, ..., m}.

where,S={S_1,S₂.....,S_m} is an m-tuple of pure strategy sets, one for each player, and F = {F_1,F₂......F_m}

In order for a game to be in normal form we assume that there is a finite set P of players, which we label {1, 2, ..., m}& Each player k in P has a finite number of pure strategies.

A set of payoffs can be considered a set of N-tuples, where N is the number of players in the game, and the cardinality of the set is equal to the total number of possible outcomes when the strategies of the players are varied. The payoff set can thus be partially ordered, where the partial ordering comes from the value of each entry in the N-tuple. while outcome is a set of moves or strategies taken by the players, or it is their payoffs resulting from the actions or strategies taken by all players. The two are complementary in that, given knowledge of the set of strategies of all players, the final state of the game is known, as are any relevant payoffs. In a game where chance or a random event is involved, the outcome is not known from only the set of strategies, but is only realized when the random event(s) are realized.

In this case,Suppose there are at least 3 possible outcomes. The expected values of the Bernoulli payoff functions u and v represent the same preferences over lotteries (and certain outcomes) if and only if there exist numbers d and c, with c>0 such that v(x) = d + cu(x).