Game Theory with Applications to Finance and Marketing

in example 2. A mixed strategy NE is defined similarly, where theequilibrium strategies are mixed strategies. 4More formally, a profile σ ∈ Σ is a (mixed strategy) NE if and only ifu i (σ) ≥ u i (s i , σ −i ), ∀i ∈ I, ∀s i ∈ S i .Observe that this definition asks us to check unilateral deviations inpure strategy only! Show that, however, it implies thatu i (σ) ≥ u i (σ ′ i, σ −i ), ∀i ∈ I, ∀σ ′ i ∈ Σ i .Show that the following is an equivalent definition for (mixed strategy)NE: a profile σ ∈ Σ is an NE if and only if for all i ∈ I, for all s i , s ′ i ∈ S i,u i (s i , σ −i ) < u i (s ′ i , σ −i) ⇒ σ i (s i ) = 0.This new equivalence condition says that if a pure strategy s i is assignedby σ i with a strictly positive probability, then s i has to be a bestresponse against σ −i .15. A single auction consists of a seller facing more than one buyer (or abuyer facing more than one seller) with some object(s) to sell (respectively,to buy), where the multiple buyers (respectively, the multiplesellers) submit bids to compete for the object(s). The object is ofprivate value if learning other bidders’ valuations for the object doesnot change one’s own valuation, and it is of common value if biddershave an (unknown) identical valuation. The four popular auction rulesare English auction, Dutch auction, the first-price sealed-bid auction,and the second-price sealed-bid auction. With an English auction, theseller-auctioneer will start with a minimum acceptable price and raisethe price gradually to attain a situation where there is exactly onebidder remains interested in the object, and in the latter situation theobject is rendered to that bidder in exchange for the price at which the4 The following interpretation may be helpful. In example 2, if player 2 expects player1 to play U, then his best response is to play R, and if player 1 correctly expects that theabove is player 2’s belief, then he cannot gain by picking an action that proves that player2’s belief is incorrect. An NE is by definition a situation where, if the two players expecttheir strategy profile, they cannot gain by making unilateral deviations.8