Mathematical Game Theory and Applications by Mazalov Vladimir;

Author:Mazalov, Vladimir;
Language: eng
Format: epub
ISBN: 9781118899632
Publisher: Wiley
Published: 2014-07-26T00:00:00+00:00

6.5 Bargaining models with incomplete information

Negotiations accompany any transactions on a market. Here participants are sellers and buyers. In recent years, such transactions employ the system of electronic tenders. There exist different mechanisms of negotiations. We begin with the double auction model proposed by K. Chatterjee and W.F. Samuelson [1983].

6.5.1 Transactions with incomplete information

Consider a two-player game with incomplete information. It engages player I (a seller) and player II (a buyer). Each player possesses private information unavailable to the opponent. Notably, the seller knows the manufacturing costs of a product (denote them by s), whereas the buyer assigns some value b to this product. These quantities are often called reservation prices. Assume that reservation prices (both for sellers and buyers) have the uniform distribution on a market. In other words, if we select randomly a seller and buyer on the market, their reservation prices s and b represent independent random variables with the uniform distribution within the interval [0, 1].

Players appear on the market and announce their prices for a product, S and B, respectively. Note that these quantities may differ from the reservation prices. Actually, we believe that S = S(s) and B = B(b)—they are some functions of the reservation prices. The transaction occurs if B ≥ S. A natural supposition claims that S(s) ≥ s and B(b) ≤ b, i.e., a seller overprices a product and a buyer underprices it. If the transaction takes place, we suppose that the negotiated price is (S(s) + B(b))/2.

In fact, players gain the difference between the reservation prices and the negotiated price: (S(s) + B(b))/2 − s (the seller) and b − (S(s) + B(b))/2 (the buyer). Recall that b and s are random variables, and we define the payoff functions as the mean values

(5.1)

and

(5.2)

The stated Bayesian game includes the functions B(b) and S(s) as the strategies of players. It seems logical that these are non-decreasing functions (the higher the seller’s costs or the buyer’s price, the greater are the offers of the players). Find a Bayesian equilibrium in the game with the payoff functions (5.1)–(5.2).

Theorem 6.11 The optimal strategies in the transaction problem have the following form:

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