Eco514-Game Theory Lecture 4: Games with Payoff Uncertainty(1 Marciano siniscalchi September 28, 1999 Introduction The vast majority of games of interest in economics, finance, political economy etc. involve some form of payoff uncertainty. A simple but interesting example is provided by auctions an object is offered for sale, and individuals are required to submit their bids in sealed envelopes. The object is then allocated to the highest bidder at a price which depends on every bid, according to some prespecified rule(e. g. a" first-price "or "second-price"rule). In many circumstances(e. g. mineral rights auctions) it is reasonable to assume that the value of the object is not known to the buyers, but that they receive some signal correlated with it. In other cases, each buyer knows the value of the object to her, but not the other players valuation We can model this situation assuming that the set of actions available to every player is the nonnegative real half-line, or some subset of it, representing allowable bids. While a profile of bids specifies the "outcome"of the auction, i. e. who receives the object and what price she pays for it, it does not specify the payoff to the winning bidder Her payoff is determined by the(uncertain) value of the object; the uncertainty may even be resolved only after the bidding game is over(e.g. when drilling or mining actually begins in the“ mineral rights”case) Thus, we need to extend our basic model of simultaneous games in order to account for payoff uncertainty The second issue is related to the solution concept(s) to apply in this setting. In the auction example, it is reasonable to assume that players will condition their bid on whatever signal they receive. In particular, in the "mineral rights"case, for any profile of bids, players cannot compute their actual payoff because of the underlying uncertainty, but they can at least compute their conditional expected payoff given their signal This implies that a solution concept for games with payoff uncertainty should specify an action(or a set of actions) for each player and for each realization of whatever signal thatEco514—Game Theory Lecture 4: Games with Payoff Uncertainty (1) Marciano Siniscalchi September 28, 1999 Introduction The vast majority of games of interest in economics, finance, political economy etc. involve some form of payoff uncertainty. A simple but interesting example is provided by auctions: an object is offered for sale, and individuals are required to submit their bids in sealed envelopes. The object is then allocated to the highest bidder at a price which depends on every bid, according to some prespecified rule (e.g. a “first-price” or “second-price” rule). In many circumstances (e.g. mineral rights auctions) it is reasonable to assume that the value of the object is not known to the buyers, but that they receive some signal correlated with it. In other cases, each buyer knows the value of the object to her, but not the other players’ valuation. We can model this situation assuming that the set of actions available to every player is the nonnegative real half-line, or some subset of it, representing allowable bids. While a profile of bids specifies the “outcome” of the auction, i.e. who receives the object and what price she pays for it, it does not specify the payoff to the winning bidder. Her payoff is determined by the (uncertain) value of the object; the uncertainty may even be resolved only after the bidding game is over (e.g. when drilling or mining actually begins, in the “mineral rights” case). Thus, we need to extend our basic model of simultaneous games in order to account for payoff uncertainty. The second issue is related to the solution concept(s) to apply in this setting. In the auction example, it is reasonable to assume that players will condition their bid on whatever signal they receive. In particular, in the “mineral rights” case, for any profile of bids, players cannot compute their actual payoff because of the underlying uncertainty, but they can at least compute their conditional expected payoff given their signal. This implies that a solution concept for games with payoff uncertainty should specify an action (or a set of actions) for each player and for each realization of whatever signal that 1