普林斯顿大学博弈论讲义10(6)

普林斯顿大学博弈论讲义3-10

chosen pro?le then becomes publicly observable.We quite simply replace individual actions with action pro?les in the de?nition of a history,and adapt the notation accordingly. Remark0.1Let A(h)={a∈ i∈P(h)A:(h,a)∈H}.Then A(h)= i∈P(h)A i(h).

The de?nition of a strategy needs minimal modi?cations:

De?nition6Fix an extensive-form gameΓwith observable actions and chance moves. Then,for every player i∈N∪{c},a strategy is a function s i:{h:i∈P(h)}→A such that,for every h such that i∈P(h),s i(h)∈A i(h).Denote by S i and S the set of strategies of Player i and the set of all strategy pro?les.

In the absence of chance moves,De?nition4applies verbatim to the new setting.You can think about how to generalize it with chance moves(we do not really wish to treat Chance as an additional player in a normal-form game,so we need to rede?ne the payo?functions in the natural way).Finally,the de?nition of Nash equilibrium requires no change.

For those of you who are used to the traditional,tree-based de?nition of an extensive game,note that you need to use information sets in order to describe games without perfect information,but with observable actions.That is,you need to use the full expressive power of the tree-based notation in order to describe what is a slight and rather natural extension of perfect-information games.1

Most games of economic interest are games with observable actions,albeit possibly with payo?uncertainty;hence,the OR notation is su?cient to deal with most applied problems (payo?uncertainty is easily added to the basic framework,as we shall see).

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