9.3. AN EQUILIBRIUM MODEL OF BANKRUPTCY 9.3.3 A strategic game Now suppose that the asset holdings of all the different categories of claimants creditors and shareholders, are common knowledge and that the preferences are common knowledge among the claimants, but not to the receiver. Instead the claimants report their cash endowments and valuations to the receiver, who then chooses a lex-efficient allocation relative to the reported valuations The claimants choose their reports to maximize their payoffs, taking as given the strategies of the other claimants and the receiver. a strategy profile is a pair of measurable functions(, 0): A-R+R+, where w(a)is agent a's reported cash holding and i(a) is his reported valuation of the equity in the re-organized firm. For each strategy profile(, i the receiver chooses a lex-efficient allocation f= (w, i), where lex-efficiency is defined relative to the reported economy rather than the true economy a technical problem in defining the game is that lex-efficient allocations are only defined up to a set of measure zero. To get around this problem, we select one version of the allocation. The allocation is symmetric or anony mous in the sense that the bundle received by an individual agent depends only on his report and the distribution of reports by other agents. Assume that the receiver chooses a market value for the firm V and a security price vector p and then gives an agent a reporting ((a), v(a)), a bundle(a, e) satisfying e+ula (a)+p 0 if i(a)<V, 0 if i(a>v. Feasibility requires w(a)sw(a). For any(V, p) it is a Nash equilibrium for agents to put w(a)=w(a) and i(a)=minv,v(a)h Proposition 3 Let(w, i)be a Nash equilibrium of the revelation game and let(v, p) be the prices defining the lear-efficient allocation implemented by the receiver. Then i(a) V for almost every a and w(a)=w(a) if u(a)>v for almost all a. Furthermore, there e.ists a Nash equilibrium in which w (a) w(a) and i(a)=min(V, u(a for almost all a If v(a)< v then agent a will receive none of the equity and so his reported valuation and wealth are(within certain bounds) indeterminate. For these agents, truth-telling is a best response, but only one of many9.3. AN EQUILIBRIUM MODEL OF BANKRUPTCY 7 9.3.3 A strategic game Now suppose that the asset holdings of all the different categories of claimants, creditors and shareholders, are common knowledge and that the preferences are common knowledge among the claimants, but not to the receiver. Instead, the claimants report their cash endowments and valuations to the receiver, who then chooses a lex-efficient allocation relative to the reported valuations. The claimants choose their reports to maximize their payoffs, taking as given the strategies of the other claimants and the receiver. A strategy profile is a pair of measurable functions ( ˆw, vˆ) : A → R+ × R+, where wˆ(a) is agent a’s reported cash holding and vˆ(a) is his reported valuation of the equity in the re-organized firm. For each strategy profile ( ˆw, vˆ) the receiver chooses a lex-efficient allocation f = Φ( ˆw, vˆ), where lex-efficiency is defined relative to the reported economy rather than the true economy. A technical problem in defining the game is that lex-efficient allocations are only defined up to a set of measure zero. To get around this problem, we select one version of the allocation. The allocation is symmetric or anony￾mous in the sense that the bundle received by an individual agent depends only on his report and the distribution of reports by other agents. Assume that the receiver chooses a market value for the firm V and a security price vector p and then gives an agent a reporting ( ˆw(a), vˆ(a)), a bundle (x, e) satisfying x + ˆv(a)e = ˆw(a) + p · d(a), e = 0 if vˆ(a) < V, x = 0 if vˆ(a) > V. Feasibility requires wˆ(a) ≤ w(a). For any (V,p) it is a Nash equilibrium for agents to put wˆ(a) = w(a) and vˆ(a) = min{V,v(a)}. Proposition 3 Let ( ˆw, vˆ) be a Nash equilibrium of the revelation game and let (V,p) be the prices defining the lex-efficient allocation implemented by the receiver. Then vˆ(a) ≤ V for almost every a and wˆ(a) = w(a) if v(a) > V for almost all a. Furthermore, there exists a Nash equilibrium in which wˆ(a) = w(a) and vˆ(a) = min{V,v(a)} for almost all a. If v(a) < V then agent a will receive none of the equity and so his reported valuation and wealth are (within certain bounds) indeterminate. For these agents, truth-telling is a best response, but only one of many