Bayesian Games: Incomplete Information

Battle of the Sexes with uncertain interestbosbayes Row loves Column, but is unsure whether Column wants to meet (type \(\theta^m\), probability \(\tfrac12\)) or avoid her (type \(\theta^a\)): meeting types have BoS payoffs; avoiding types get the BoS payoffs with Column’s coordination payoffs flipped to miscoordination. Solving type-by-type: the avoiding type has a dominant tendency to mismatch, and Row optimally goes to her favorite event regardless; the meeting type follows Row. Working through the best responses gives a strict BNE: Row plays \(B\); Column plays \(B\) if \(\theta^m\), \(F\) if \(\theta^a\). The point: equilibrium play of every type must be specified, even types that do not occur in the realized play.

Cournot with unknown costscournotbayes Firm 1’s cost is \(c\) (common knowledge); firm 2’s cost is \(c_L\) w.p. \(\mu\) or \(c_H\) w.p. \(1 - \mu\), privately known. Demand \(P = a - q_1 - q_2\). Each firm-2 type sets \(q_2(c_2) = \frac{a - c_2 - q_1}{2}\); firm 1 best-responds to the expected output \(\bar q_2 = \mu q_2(c_L) + (1-\mu) q_2(c_H)\). Solving the three linear equations gives, with \(\bar c_2 = \mu c_L + (1-\mu) c_H\): \[q_1^* = \frac{a - 2c + \bar c_2}{3}, \qquad q_2^*(c_2) = \frac{a - 2 c_2 + c}{3} + \frac{c_2 - \bar{c}_2}{6}.\] The high-cost type produces more than it would under complete information (its rival, not knowing costs are high, produces less aggressively than it otherwise would). Information asymmetries change strategic behavior even for the uninformed party.

Second-price (Vickrey) auctionvickrey Highest bidder wins, pays the second-highest bid. As shown in Example [exa:spapreview], bidding \(b_i = v_i\) is weakly dominant. Truthful bidding requires no knowledge of \(F\), of \(n\), or of others’ rationality—an ideal of robust mechanism design. Revenue: the expected second-highest valuation.

First-price auctionfirstprice Highest bidder wins, pays her own bid. Now bidding \(v_i\) guarantees zero profit; optimal bids shade below the valuation, trading off win probability against margin. Look for a symmetric increasing equilibrium \(b(v)\). Bidder \(i\) with value \(v\) choosing bid \(b(z)\) (imitating type \(z\)) wins iff all rivals’ values are below \(z\), probability \(F(z)^{n-1}\); expected profit \(\pi(z) = F(z)^{n-1} (v - b(z))\). The FOC at \(z = v\) plus the boundary condition \(b(0)=0\) yields \[b(v) \;=\; v - \frac{\int_0^v F(x)^{n-1}\,dx}{F(v)^{n-1}} \;=\; \E\big[\max_{j \neq i} v_j \,\big|\, \max_{j \neq i} v_j < v\big].\] With uniform values on \([0,1]\): \(b(v) = \frac{n-1}{n} v\). You bid the expected highest rival value conditional on winning—shading exactly to the point where you’d tie with your strongest rival.

The winner’s cursewinnerscurse An oil tract has the same unknown value \(V\) to all bidders; each observes a noisy private signal \(s_i = V + \varepsilon_i\). If everyone bids their unbiased estimate, the winner is the bidder with the largest positive error: winning is bad news. Rational bidders shade for this selection effect, bidding as if their signal were the highest—i.e., conditioning on the event of winning. Inexperienced bidders systematically overpay (documented in offshore oil leases, book publishing advances, free agency in sports, and corporate takeovers).

Akerlof’s lemons marketlemons A seller knows her car’s quality \(q \sim U[0,1]\); the buyer values quality \(q\) at \(\tfrac32 q\), the seller at \(q\). At price \(p\), only sellers with \(q \le p\) sell; average quality traded is \(p/2\), worth \(\tfrac34 p < p\) to the buyer. The market unravels: only \(q = 0\) trades in equilibrium, though every car has positive gains from trade. Asymmetric information alone can destroy a market—the founding result of information economics (Akerlof 1970, Nobel 2001), and the reason for warranties, inspections, and certification (signaling: next chapter).

Currency attack (Morris–Shin 1998, sketch)globalgame Speculators decide whether to attack a currency peg; attacking costs \(t\) and pays 1 if the peg breaks, which happens if enough speculators attack relative to fundamentals \(\theta\). With \(\theta\) common knowledge, there is a range of \(\theta\) with two equilibria (attack/no attack)—self-fulfilling crises. Now let each speculator observe \(\theta\) with tiny idiosyncratic noise. Iterated dominance from the extreme regions (very bad \(\theta\): attacking dominant; very good: abstaining dominant) propagates inward and selects a unique equilibrium: attack iff the signal is below a cutoff \(\theta^*\). Vanishingly small uncertainty destroys the multiplicity, and the selected cutoff corresponds to the risk-dominant behavior. Global games are now a standard tool for bank runs, debt crises, and regime change.