player may observe However, several subtle issues arise regarding the interpretation of "signals. If the latter can be literally take en to be phy I observations(e. g. samples or geological then it makes sense to assume that players share a common probabilistic description of the underlying uncertain quantities and their correlation with each player's signal. Such description could even be said to be part of the formulation of the game If, however, signals do not have a readily available physical interpretation(consider the value, i. e, the maximum amount one is willing to pay for a painting, which presumably includes a subjective element), then it does not make much sense to postulate that players agree on any given probabilistic model of the underlying uncertain quantities-let alone make such description part of the formulation of the game. Rather, a specification of players beliefs about the underlying uncertainties should be considered as part of the solution concept Further observations will be in order once we specify a formal model of games with payoff uncertainty Games with Payoff uncertainty I follow OR (with minimal expository deviations) and begin by providing a rather general definition Definition 1 A(finite)normal-form game with payoff uncertainty is a tuple=(N, Q, (Ai, ui, TiieN) where N is a finite set of players, Q2 is a set of states of the world, and for each i E N, Ai is a set of actions, u;: Ai x A-ix Q-r is Player i' s payoff function, and Ti is a partition of Q, referred to as Player i's type partition It may be helpful to relate normal-form games with payoff uncertainty with the decision- eoretic framework introduced in Lecture 1. Taking the perspective of Player i, the set of states of nature for i is Q2= A-i X Q; the set of acts is(isomorphic to) Ai, with each act ai E Ai mapping a state(a-i, w)En to the outcome determined by the tuple(ai, a-i, w) finally, Player i's preferences over outcomes are represented by u Thus, Player i is not only uncertain about her opponents'actions: she is also uncertain as to the prevailing state of the world w E Q. In keeping with our general framework, a solution concept should specify a probability distribution over whatever a player is uncertain about hence, it must also specify her beliefs about the prevailing state of the world Note: the usual textbook"presentation of the situations referred to herein uses the terminology "games with incomplete information, and includes a specification of a common prior over the set of states of the world as part of the description of the strategic situation We use the somewhat nonstandard term"payoff uncertainty'"to emphasize that we do not wish to regard probabilities over states of the world as part of the model, but rather as partplayer may observe. However, several subtle issues arise regarding the interpretation of “signals.” If the latter can be literally taken to be physical observations (e.g. samples or geological surveys), then it makes sense to assume that players share a common probabilistic description of the underlying uncertain quantities and their correlation with each player’s signal. Such description could even be said to be part of the formulation of the game. If, however, signals do not have a readily available physical interpretation (consider the value, i.e, the maximum amount one is willing to pay for a painting, which presumably includes a subjective element), then it does not make much sense to postulate that players agree on any given probabilistic model of the underlying uncertain quantities—let alone make such description part of the formulation of the game. Rather, a specification of players’ beliefs about the underlying uncertainties should be considered as part of the solution concept. Further observations will be in order once we specify a formal model of games with payoff uncertainty. Games with Payoff Uncertainty I follow OR (with minimal expository deviations) and begin by providing a rather general definition. Definition 1 A (finite) normal-form game with payoff uncertainty is a tuple G = (N, Ω,(Ai , ui , Ti)i∈N ), where N is a finite set of players, Ω is a set of states of the world, and for each i ∈ N, Ai is a set of actions, ui : Ai × A−i × Ω → R is Player i’s payoff function, and Ti is a partition of Ω, referred to as Player i’s type partition. It may be helpful to relate normal-form games with payoff uncertainty with the decision￾theoretic framework introduced in Lecture 1. Taking the perspective of Player i, the set of states of nature for i is Ω = ˜ A−i × Ω; the set of acts is (isomorphic to) Ai , with each act ai ∈ Ai mapping a state (a−i , ω) ∈ Ω to the outcome determined by the tuple ˜ (ai , a−i , ω); finally, Player i’s preferences over outcomes are represented by ui . Thus, Player i is not only uncertain about her opponents’ actions: she is also uncertain as to the prevailing state of the world ω ∈ Ω. In keeping with our general framework, a solution concept should specify a probability distribution over whatever a player is uncertain about; hence, it must also specify her beliefs about the prevailing state of the world. Note: the usual “textbook” presentation of the situations referred to herein uses the terminology “games with incomplete information, and includes a specification of a common prior over the set of states of the world as part of the description of the strategic situation. We use the somewhat nonstandard term “payoff uncertainty” to emphasize that we do not wish to regard probabilities over states of the world as part of the model, but rather as part 2