Avoidance and Mitigation of Public Harm

Transcription

1 Avoidance and Mitigation of Public Harm Bruno Salcedo Bruno Sultanum Ruilin Zhou The Pennsylvania State University Abstract This paper studies the optimal design of a liability sharing arrangement as an infinitely repeated game. We construct a schematic, non-cooperative, 2-player model where an active agent can take a costly, unobservable action to try to avert a crisis, and then when a crisis occurs, both agents decide how much to contribute mitigating it. The model is able to generate any combination of avoidance/mitigation patterns as static Nash equilibrium. We then consider perfect public equilibrium PPE of the infinitely repeated game. When the avoidance cost is too high relative to the expected loss of crisis for the active agent, the first best is not a static Nash equilibrium of the stage game and cannot be implemented as a PPE of the repeated game. For this case, we show that at any Pareto efficient PPE, the activeagent shirks infinitely often, and when crisis happens, the active agent is bailed out infinitely often. Keywords bailouts infinitely repeated game moral hazard JEL classification C73 D82 1. Introduction This paper studies the optimal design of an arrangement for sharing the costs of mitigating bad outcomes in an infinitely repeated game. When a bad outcome causes loss to several players, it is a public bad, and cost sharing is necessary to induce the socially optimal level of mitigation ex post. However, if it is efficient ex ante for some player to make unobservable, costly investment to avert bad outcomes, then cost sharing of mitigation might undermine that player s incentive to make the costly investment. The model to be studied here is proposed as a schematic model of ongoing relationships among parties for example, countries in the Euro zone today that jointly suffer the adverse effects of financial crises We thank Ed Green and V. Bhaskar for helpful discussion and comments. addresses: bruno@psu.edu, sultanum@psu.edu, rzhou@psu.edu 1

2 that can result from imprudent behavior that is, by shirking on investment in avoidance by a member of the group. In episodes of financial failure, a bailout occurs when one institution or country experiences a situation that causes direct harm there, and when other institutions are motivated to contribute to the cost of cleaning up the situation for example, by recapitalizing a bank because they suffer externalities from the situation. Historically, in episodes of financial turmoil, some troubled institutions have been bailed out and others have not been. As a result, the fate of these troubled parties ranges from complete failure/bankruptcy to full recovery. 1 A good positive theory needs to account for this very wide spectrum of outcomes. In view of the heterogeneity of outcomes and of the tension between ex-ante and ex-post efficiency discussed above, questions about economic efficiency are especially salient in this context. We construct a schematic, non-cooperative, 2-player model where an active agent takes costly, unobservable action to try to avert a crisis, and whenever a crisis occurs, both agents decide how much to contribute mitigating it. This is a moral hazard problem caused by the potential action taken by the active agent and the implied negative externality on the passive agent. We study the Nash equilibrium of the one-shot game. The model is able to generate any combination of avoidance/mitigation patterns as static Nash equilibrium. We then consider the infinite repetition of the previous game, and study the perfect public equilibrium PPE of the repeated game. When the avoidance cost is not too high, avoidance/mitigation, which achieves the lowest expected discounted total cost, can be easily implemented as a PPE since it is a repetition of the static Nash equilibrium. When the avoidance cost is too high relative to the expected loss of crisis for the active agent, avoidance/mitigation is no longer a static Nash equilibrium of the stage game, and also cannot be implemented as a PPE of the repeated game. This is because in order to induce the active agent to take the costly avoidance action, the expected mitigation cost for him must be even higher. With such high duo costs, the active agent is better off doing nothing. In order to compensate the active player for taking the avoidance action soemtimes, he has to be allowed to shirk other times, which would increase the incidence of crisis, and be bailed out pay less than his share of the mitigation cost when crisis happens. Based on this intuition, we show that at any Pareto efficient PPE, the active agent shirks infinitely often, and when crisis happens, the active agent is bailed out infinitely often. As a result, crisis occurs more 1 For example, while majority of US companies sink or float on their own, US government has consistently bailed out large corporations in auto-mobile industry such as GM and Chrysler. Among financial institutions, the Federal Reserve Bank gave a lot of assistance in recent economic crisis in the form of loans and guarantees to large banks like Citigroup and Bank of America, while letting other large financial institutions such as Lehman Brothers and Washington Mutual fail. Among sovereign countries, US government helped Mexico to survive the 1994 Tequila crisis, while many countries suffered huge losses during the 1997 Asian financial crisis with little help from the IMF. The outcome of current ongoing Euro area crisis is yet to be seen with whether and how much more bailout should be given to Greece, Italy, Spain and all other potential problem countries. 2

3 often compared to the first-best outcome. We demonstrate this kind of equilibrium with a numerical example, at which the first best of avoidance/mitigation cannot be supported as a PPE. We show that a particular PPE, where the active agent shirks sometimes and is bailed out other times, yields a small welfare loss relative to the first best. This result is consistent with the real-world observation that bailout happens sometimes, but not at other times. Such a phenomenan is not due to agents being indifferent, but a delibrate arrangement of using bailout to incentivize good behavior as often as possibile. 2. The one-shot game There are two agents, the active agent 1 and the passive agent 2 active versus passive reflects that player 2 will have no role in the first period of the two-period stage game.. There are two periods. In period 1, the active agent chooses whether to take a costly action to avert a crisis, a A = {0, 1}. The cost of avoidance is d > 0, and not taking the action costs nothing. Agent 1 s first period action a affects the distribution of the state in period 2, ξ X = {0, 1}. If ξ = 1, a crisis occurs, and if ξ = 0, there is no crisis. The probability of a crisis in period 2 is π 1 0, 1 if agent 1 takes the the avoidance action a = 1, and π 0 0, 1 otherwise a = 0. That is, Prξ = 1 a = π a, Prξ = 0 a = 1 π a. 1 We assume that π 1 < π 0 so that taking the avoidance action reduces the probability of the crisis occurring. In period 2, the state ξ is realized. If there is a crisis ξ = 1, the two agents simultaneously decide how much to contribute to mitigate it. Let m i denote agent i s contribution, m i M =R +. The crisis is mitigated if the total contribution of the two agents m 1 + m 2 is no less than 1. Otherwise, each agent i suffers a loss c i due to the crisis in addition to the cost of his contribution m i. If there is no crisis, the two agents incur no loss due to crisis, only mitigation cost if they choose positive contributions, which would be completely wasted since there is no crisis to mitigate and contributions cannot be consumed by either agent. Figure 1 summarizes the structure of the game. We are interested in studying the case where mitigation after a crisis is ex-post efficient. So we impose the following assumption. Assumption 1 For i = 1, 2, c i 0, 1, and c 1 + c 2 > 1. This assumption implies that neither agent alone will be willing to mitigate the crisis, but together they should since the cost of mitigation, 1, is less than the total loss if the crisis is not mitigated, c 1 + c 2. 3

4 1 a = 0 a = 1 Nature Nature π 0 1 π 0 π 1 1 π m 1 m 1 m 1 m m 2 m 2 m 2 m 2 m1 c 1 I {m1 +m 2 <1} m 2 c 2 I {m1 +m 2 <1} m 1 m 2 m1 c 1 I {m1 +m 2 <1} d m 2 c 2 I {m1 +m 2 <1} m 1 d m 2 Figure 1 The one-shot game 2.1. Equilibrium The structure of the game allows us to restrict attention to pure and public strategies without loss of generality. Players have no use for randomization of the mitigation action because they are risk neutral and M is convex. The active player has no reason to randomize the avoidance action because it is the only action chosen at the first stage. The only private information is the avoidance action of the active player, and he finds no benefit from conditioning his choice on this information. We thus solve for the pure-strategy, public-perfect Nash equilibria of this two-period game between the two agents. Consider period 2 first. Let u i ξ, m 1, m 2 denote agent i s period-2 utility when the state is ξ, and the two agents contributions are m 1 and m 2. Denote agent i s period-2 strategy by m i ξ, m i : X M. When there is a crisis ξ = 1, the mitigation game is played. Agent i s payoff is u i 1, m 1, m 2 = m i c i I {m1 +m 2 <1} 2 Let ρ i m i denote agent i s response in a crisis state when the other agent s contribution is m i, ρ i : M M. Then agent i s optimal response correspondence is 0 if c i < 1 m i ρ i m i = [0, 1 m i ] if c i = 1 m i 3 1 m i otherwise The fixed points of the product correspondence of the two agents optimal response correspondences determines the Nash equilibria of the period-2 mitigation game, m 1 1, m

5 By assumption 1, there are two types of period-2 equilibria: No-mitigation: neither agent contributes anything, m o 1 1 = mo 2 1 = 0. The period- 2 utility of agent i is then u i 1, m o 1 1, mo 2 1 = c i. Mitigation: the two agents jointly contribute 1 unit to mitigate the crisis, m b 11 [1 c 2, c 1 ], m b 2 1 = 1 mb 1 1. The period-2 utility of agent i in this equilibrium is then u i 1, m b 11, m b 21 = m b i1. When there is no crisis ξ = 0, there is no loss due to crisis, so period-2 utility of agent i is u i 0, m 1, m 2 = m i 4 The optimal action is not to contribute anything. So at equilibrium, m 1 0 = m 2 0 = 0, and period-2 utility of agent i is u i 0, m 1 0, m 2 0 = 0. For the remainder of this section we maintain the restriction m i 0 = 0. Now consider agent 1 s period-1 problem of whether to take the avoidance action, a A, assuming that the two agents are going to play their equilibrium strategies in period 2. Let v i a, m 1, m 2 denote the expected value of agent i at the beginning of period 1 if agent 1 takes action a in period 1 and the strategy profile in period 2 is m 1 ξ, m 2 ξ ξ X. v 1 a, m 1, m 2 = ξ X Prξ au 1 ξ, m 1 ξ, m 2 ξ ad 5 v 2 a, m 1, m 2 = ξ X Prξ au 2 ξ, m 1 ξ, m 2 ξ 6 Agent 1 s optimal period-1 action a depends on which of the period-2 equilibrium is to be played in case of crisis. If no-mitigation equilibrium m o 1 1, mo 2 1 is anticipated, then π 0 c v 1 a, m o 1, m o 1 if a = 0 2 = d π 1 c 1 if a = 1 and the optimal action a o is 1 if c 1 ˆd a o = 0 otherwise 7 where ˆd is a rescale of the avoidance cost d adjusting by its impact on the probability of a crisis, ˆd d π 0 π1. 8 5

6 By a similar argument, if the mitigation equilibrium m b 11, m b 21 is anticipated, then the optimal action a b is 1 if m b a b 1 1 ˆd = 0 otherwise Putting the two-period analysis together, Table 1 summarizes all equilibria of the oneshot game in different parameter regions that depend on the adjusted cost of avoidance ˆd. Note that, by Assumption 1, 0 < 1 c 2 < c 1 < 1, none of the parameter region is empty. The table includes only equilibrium range of agent 1 s contribution m 1 1. For equilibrium with mitigation where m 1 1 takes value in an interval, m 2 1 = 1 m 1 1. For equilibrium without mitigation, m 2 1 = 0. 2 The table indicates two important results. First, regardless of the parameter region, both mitigation and no-mitigation equilibrium always coexist. Second, every possible combination of avoidance and mitigation occurs in at least one parameter region. 9 Table 1 Equilibria of the one-shot game parameter region a m 1 1 mitigation/not ex-ante cost ˆd 1 c 2 < c 1 < 1 1 c 2 < ˆd c 1 < 1 c 1 < ˆd 1 [1 c 2, c 1 ] yes d + π no d + π 1 c 1 + c 2 [ ] 1 ˆd, c1 yes d + π 1 [ 0 1 c2, ˆd ] yes π no d + π 1 c 1 + c 2 0 [1 c 2, c 1 ] yes π no π 0 c 1 + c Welfare The last column of Table 1 lists the ex-ante expected total cost C at each equilibrium, which includes the cost of taking avoidance action, the expected cost of mitigation and/or the loss due to crisis if there is no mitigation: C = ad + 1 aπ 0 + aπ 1 [ m 1 + m 2 + c 1 + c 2 I {m1 +m 2 <1} 2 Note that in each case, there is always a continuum of mitigation equilibrium where the two agents split of 1 unit of contribution varies over an interval. This is due to the assumption that the mitigation game in period 2 in case of crisis is a simultaneous-move game. Other assumptions in the game form, such as the two agents make contributions sequentially, would yield a smaller set of equilibria. ] 10 6

7 We use this cost to discuss the ex-ante welfare ranking of different coexisting equilibria. An action profile is said to ex-ante dominate another one, if it has a lower expected total cost. We have assumed that when there is a crisis, it is less costly to mitigate than to suffer the consequence c 1 + c 2 > 1, i.e. ex-post mitigation dominates no-mitigation. Table 1 shows that when the cost of avoidance is low ˆd 1 c 2, agent 1 takes the avoidance action in both mitigation and no-mitigation equilibria. When the cost of avoidance is high ˆd > c 1, agent 1 does not take the action in either type of equilibria. In either case, the assumption that mitigation dominates ex-post implies that it also dominates ex-ante, because agent 1 s period-1 action is the same in both mitigation/no-mitigation equilibria. The intermediate case is more complicated. There are three types of equilibria in this region: i avoidance/mitigation, ii no-avoidance/mitigation and iii avoidance/nomitigation. Since here d + π 1 < π 0 since ˆd < 1, Assumption 1 implies that i dominates ex-ante both ii and iii, but there is no clear welfare ranking of cases ii and iii ex-ante. 3. The repeated game Time is discrete, dates are indexed by t = 1, 2,.... At each date t, the two agents, who have a common discount factor δ 0, 1, play the two-period stage game described above. At the beginning of date t a payoff-irrelevant public signal sunspot θ t Θ = [0, 1] is observed, these signals are i.i.d. U[0, 1]. Agent 1 then decides to take the avoidance action or not, a t A. The state ξ t X is then drawn independently according to the distribution Prξ t a t, as given in 1. Both agents observe the realization of state variable ξ t before deciding simultaneously their respective contributions, m t1, m t2 M M. Denote the date-t public information by h t = θ t, ξ t, m t1, m t2 H Θ X M M, and let h t = h 1,..., h t 1 H t 1 where H 0 = and H t = H t 1 H. The public history at stage 1 of date t is h t, θ t H t 1 Θ. The public history at stage 2 of date t is h t, θ t, ξ t H t 1 Θ X. We focus on perfect Bayesian equilibria where both agents play pure and public strategies. The structure of the game allows us to do this without loss of generality, as shown in Proposition 2. Formally, a public strategy for agent 1 is a sequence of measurable functions σ 1 = {α t, µ t1 } t=1, and for agent 2 is a sequence σ 2 = {µ t2 } t=1, where α 1 : Θ A, µ 1i : Θ X M, and for t > 1, α t : H t 1 Θ A, µ ti : H t 1 Θ X M for i = 1, 2. That is, at date t, given a public history h t and public signal θ t, agent 1 takes the avoidance action α t h t, θ t. Then, given ξ t, the two agents contribute µ t1 h t, θ t, ξ t, µ t2 h t, θ t, ξ t to mitigate crises. Associated with strategy profile σ = σ 1, σ 2 is a stochastic stream of payoff vectors. The 7

8 expected present discounted value of this stream is denoted by V σ = V 1 σ, V 2 σ, [ ] V i σ = 1 δe δ t 1 v i αt, µ 1t, µ 2t 11 t=1 where v i is the single period payoff function of agent i given by 5 6. For any public history h t, let σ h t = α h t, µ 1 h t, µ 2 h t denote the strategy profile induced by σ after the history h t. Definition 1 A public strategy profile σ = α, µ 1, µ 2 is a perfect public equilibrium PPE if and only if for any t 1: i For any h t H t 1 and for any other public strategy σ 1: V 1 σ 1 h t, σ 2 h t V 1 σ 1, σ 2 h t 12 ii For any h = h t, θ t, ξ t H t 1 Θ X and for any pair of public strategies σ 1 and σ 2 : 1 δu 1 ξ t, µ t1 h, µ t2 h + δv 1 σ1 h, σ 2 h 1 δu 1 ξ t, µ t1h, µ t2h + δv 1 σ 1 h, σ 2 1 h 1 1 δu 2 ξ t, µ t1 h, µ t2 h + δv 2 σ1 h, σ 2 h 1 δu 2 ξ t, µ t1h, µ t2h + δv 1 σ 1 h, σ 2 2 h 2 where h = h, µ t1 h, µ t2 h, h 1 = h, µ t1 h, µ t2 h, and h 2 = h, µ t1 h, µ t2 h. Condition 12 states that at the beginning of period t, agent 1 s continuation strategy α h t, µ 1 h t is optimal from period t onward after any history h t. Conditions 13 and 14 state that at period t, after the state ξ t is realized, both agents contribution strategies at period t, µ t1h, µ t2h and continuation strategies α h, µ 1 h, µ 2 h from period t + 1 onward are optimal, for any history h = h t, θ t, ξ t. Let V denote the set of PPE payoff vectors, V = { V σ σ is a PPE } This set is non-empty given that a PPE always exists since unconditional repetition of a static Nash equilibrium of the stage game is a PPE, and we have shown that the stage game always has equilibria

9 3.1. APS decomposition With agents playing public strategies only, this repeated game has a recursive structure. After an arbitrary history, the continuation strategy profile of a PPE is an equilibrium profile of the original game. The standard way to characterize the set of PPE values is to use the self-generation procedure introduced in Abreu et al The rest of this section is devoted to establishing an analogous procedure and showing that our restriction to pure and public strategies is without loss of generality. A reader who is not interested in these technical results might skip to the economic analysis of the rest of the paper. The model here differs from APS in three aspects. The stage game is a multi-stage game. We allow for public randomization but exclude individual mixed strategies. Our game is not purely a game with perfect monitoring actions are perfectly observable, nor one purely with imperfect public monitoring agents observe only noisy information of past actions. It is a mixture of the two: at any date t, agent 1 s avoidance action a t is his private information, agent 2 sees only the a t -induced realization of state ξ t, yet both agents observe each other s contribution m 1t and m 2t. Nonetheless, the self-generation procedure can be used to generate the set of PPE values V. The expected payoff for player i = 1, 2 of action profile s = a, m 1, m 2 when the continuation values are given by w : X M 2 R 2 is 3 g i s, w 1 δv i s + δ ξ X Prξ a w i ξ, m1 ξ, m 2 ξ 15 Definition 2 An action profile s = a, m 1, m 2 together with a continuation payoff function w : X M 2 R 2 is admissible with respect to a set W R 2 if and only if: i for all ξ X, wξ, m 1 ξ, m 2 ξ W. ii a arg max a A ga, m 1, m 2, w iii For any ξ X: m 1ξ arg max m 1 M 1 δu 1 ξ, a, m 1, m 2ξ + δw 1 ξ, m 1, m 2ξ m 2ξ arg max1 δu 2 ξ, a, m 1ξ, m 2 + δw2 ξ, m 1 ξ, m 2 m 2 M Condition i says that the continuation payoff takes value in the W set. Conditions ii and iii require that the action profile a, m 1, m 2 is a Nash equilibrium if the two agents payoff function is given by g i. 3 The sunspots do not appear explicitly the sequential formulation. The action profile s specifies the pure actions chosen after θ t is realized, and the continuation value w is taken to be the average continuation values integrating over θ t+1. Public randomization enters implicitly in the the convex hull operation in 17. 9

10 In our setting with public randomization, the adequate self-generating operator is the one in Cronshaw and Luenberger For any set W R 2 and every action profile s = a, m 1, m 2 define: B s W = { v W w : X M 2 W such that } s, w is admissible w.r.t. W and v = gs, w 16 and: BW = co B s W s S 17 A set W R 2 is said to be self-generating if and only if W BW. The following proposition states that using this notion of self-generation, the characterization of APS applies to our setting. The proofs are in appendix B. Proposition 1 V is compact and convex and it is the largest-self generating set. Furthermore, if W is bounded and V W then B n W V. We are now in a position to prove our claim about the restriction to pure-public strategies being without loss of generality. One could extend the definition of equilibrium in the obvious way to allow for mixed strategies that depend on private information. The following proposition states that the set of equilibrium payoffs would not change. The reason for this is because the new set would be self-generating in the original sense, and thus it would be contained in V. Proposition 2 The set of equilibrium payoffs would remain unchanged if we allowed player 1 to use private strategies and we allowed both players to use mixed strategies. 4. The case when avoidance is not possible In the first two cases form table 1, when ˆd c 1, the minimum total cost can be achieved by static Nash equilibria which combine avoidance/mitigation. On the other hand, if ˆd < 1, then avoidance is socially efficient. In this case efficiency is once again achieved by repetition of static Nash, now of the form no-avoidance/mitigation. The interesting case of the repeated game is when play avoidance is not a static Nash equilibrium even tough its cost is lower than its total benefit: 10

11 Assumption 2 c 1 < ˆd < 1. Under assumption 2 it is socially optimal to avoid the crisis because ˆd < 1, but agent 1 does not take the avoidance action in any equilibrium of the static game since c 1 < ˆd. In this situation, can infinite repetition of the stage game give rise to sufficient incentive for agent 1 to take the avoidance action, despite the fact that he can cheat undetected, so that the lowest ex-ante expected discounted lifetime total cost can be achieved? In other words, can avoidance/mitigation be sustained at some PPE of the repeated game? Unfortunately the answer is negative. Proposition 3 There is no PPE in which the avoidance action is taken at every period. The reason that full avoidance cannot be supported as an equilibrium when the cost of avoidance is too high is quite informative. The expected discounted value for the active agent V is a mixture of his expected value if there is crisis V 1 and his expected value if there is no crisis V 0. The only way to generate incentives for avoidance is to guarantee that V 0 should be sufficiently greater than V 1. On the other hand, the mitigation incentive constraints imply that V 1 cannot be too low. As a result, whenever agent 1 chooses a = 1 and there is no crisis, his continuation value must increase at least by a fixed amount. Therefore, if there is no crisis for a sufficiently long period, the implied continuation value stop being feasible. 4 Since the problem is that the avoidance cost is too high and agent 1 cannot induced to pay it every period alone, the solution seems to be that agent 2 should help to pay it. If the two agents have transferable utility, we can investigate the scheme that agent 2 subsidize agent 1 in period where crisis occurs. 5 However, we have made the assumption that each agent s contribution can be used only to mitigate the crisis, the only way for agent 2 to compensate agent 1 is by paying more in mitigation cost sometimes when crisis occur. That is, when the avoidance cost is too high for agent 1, if he can somehow be compensated on mitigation contribution, then he may be induced to take the avoidance action, at least in some periods. When this happens, we call it a bailout. Definition 3 A bailout is the situation where crisis occurs and the active party agent 1 pays less than his crisis loss c 1 in mitigation. In other words, we say that there is a bailout if µ 1t h t, 1 < c 1 and µ 1t h t, 1 + µ 2t h t, 1 1. Bailouts are the only form of compensation available. And if agent 1 is not compensated 4 It is crucial for this proposition that the avoidance action is not observable. Hence, this is a result of moral hazard and not of the structure of the payoffs. If we assume that a is observable, grim trigger d+π strategies supports avoidance in every period if δ 1 1 c 2 π 0 c 1 d+π 1 1 c 2 π 0 c 1+π 1 d+π 0 c 1+c 2 π 1 0, 1. 5 We can show that if subsidy is allowed agent 2 directly pay agent 1 in consumption goods, if both agents are sufficiently patient, the first best of avoidance/mitigation can be achieved. 11

12 then he has no reason to choose avoidance. It follows that bailouts are necessary to induce avoidance. Proposition 4 In any PPE where avoidance actions is taken with positive probability there are bailouts with positive probability Constrained Pareto efficiency There are a lot of equilibria in the static one-shot game. The set of PPE of the infinitely repeated game is even larger. Many of the equilibria are Pareto dominated. One can define the Pareto frontier between the two agents, obtained by varying agent 1 s achievable expected discounted utility while maximizing agent 2 s expected discounted utility among all PPEs. A first observation is that mitigation is required for Pareto efficiency. Proposition 5 In any constrained Pareto efficient PPE there is always mitigation after a crisis, that is µ 1t h t, 1 + µ 2t h t, 1 1 almost surely along the equilibrium path. Proposition 4 shows that bailouts are necessary in order to support avoidance actions, but it does not tell anything about efficiency. That is, in order to support a particular point on the Pareto frontier, is avoidance necessary? Assumption 2 implies only that more avoidance reduces total expected cost, but potentially it could happen at a cost for some player. Besides, there might not exist a PPE where avoidance is played at all. Proposition 6 shows that whenever is possible to support avoidance then any Pareto efficient equilibrium has avoidance and bailouts infinitely often. Proposition 6 There exists some δ 0 0, 1 such that: i If δ < δ 0, then every PPE and therefore every constrained Pareto efficient PPE has avoidance played with probability zero at all periods. ii If δ > δ 0, then every constrained Pareto efficient PPE has avoidance played infinitely often and bailouts happens infinitely often Numerical example with automaton In what follows, we provide an example in the form of an automaton where avoidance takes place at some and not at other time. Consider the set of parameters δ = 0.95, π 1 = 0.2, π 0 = 0.9, d = 0.5, c 1 = 0.6, c 2 =

13 Note that with this set of parameters, the adjusted avoidance cost is higher than the cost for agent 1 and avoidance is socially efficient, c 1 = 0.6 < ˆd < 1. The first best has an expected total cost of 0.7, but by Proposition 3 this is not obtainable in any PPE. In contrast, the first best could be sustained as a PPE if the avoidance action was observable. The lower total cost that can be obtained in static Nash equilibria is 0.9, which implies a welfare loss relative to the first best of 28.57%. a m ω 1 : 1 m ω 2 : a m 1 m Crisis No-Crisis ω 3 : a m 1 m a m ω 4 : 1 m Figure 2 Automaton Figure 2 describes a PPE that takes the form of an automaton. In state ω 1 the strategy profile is a, m 1, m 2 = 1, 0.61, If there is a crisis, the state remains to be ω 1. If there is no crisis then players stay in ω 1 with probability 0.82 and go to ω 2 with probability In ω 2 the strategy profile is 1, 0, 1. If there is a crisis then the state goes back to ω 1. If there is no crisis then players stay in ω 2 with probability 0.76 and go to o 3 with probability In ω 3 state the strategy profile is 1, 0, 1. If there is a crisis then the state goes back to ω 2. If there is no crisis then players stay in ω 3 with probability 0.76 and go to ω 4 with probability In ω 4 state the strategy profile is 0, 0, 1. If there is a crisis then players stay in ω 4 with probability 0.32 and go to ω 3 with probability If there is no crisis then players stay in ω 4 with probability 0.46 and go to ω 3 with probability There is a fifth state ω 5 that is not in the figure with strategy profile of non-avoidance/no-mitigation. This state is out of the equilibrium path and works as a punishment in case of a detectable deviation. 13

14 Table 2 Summary statistics of the sample PPE State ω Invariant distribution u 1 ω u 2 ω V 1 ω V 2 ω Welfare Welfare loss ω % ω % ω % ω % LRA % Notes: LRA refers to the long run averages, which correspond to the expected values evaluated using the invariant distribution. u i ω denotes agent i s expected payoff for the current period when the state is ω. V i ω denotes agent i s total discounted expected payoff when the current state is ω. Although this PPE may not be in the Pareto frontier, it provides a lower bound on what can be achieved by a constrained efficient allocation. In state ω 1, the expected discounted total cost is 0.709, which is only 1.29% greater than the minimum feasible one 0.7. Furthermore, in states ω 2, ω 3 and ω 4, agent 1 contributes m 0 = 0 to mitigate the crisis, rather than 0.6 his loss if there is no mitigation. By our earlier definition, this is a bailout. On average, crisis occurs 25.6% of the time, compared to 20% of the time at the first best. When crisis does happen, bailout occurs 70.3% of the time. References Abreu, D., Pearce, D., and Stacchetti, E Toward a theory of discounted repeated games with imperfect monitoring. Econometrica, 585: Admati, A. R. and Pfleiderer, P Robust financial contracting and the role of venture capitalists. The Journal of Finance, 492: Barry, C., Muscarella, C., Peavy, J., and Vetsuypens, M The role of venture capital in the creation of public companies: Evidence from the going-public process. Journal of Financial Economics, 272: Bernhardt, D., Hollifield, B., and Hughson, E Throwing good money after bad. Cornelli, F. and Yosha, O Stage financing and the role of convertible securities. The Review of Economic Studies, 701:1 32. Cronshaw, M. B. and Luenberger, D. G Strongly symmetric subgame perfect equilibria in infinitely repeated games with perfect monitoring and discounting. Games and Economic Behavior, 62: Diamond, D. W Financial intermediation and delegated monitoring. The Review of Economic Studies, 513: Diamond, D. W Monitoring and reputation: The choice between bank loans and directly placed debt. Journal of Political Economy, pages Friedman, B., Kuttner, K., Bernanke, B., and Gertler, M Economic activity and the short-term credit markets: An analysis of prices and quantities. Brookings Papers on Economic Activity, 19932:

15 Gale, D. and Hellwig, M Incentive-compatible debt contracts: The one-period problem. The Review of Economic Studies, 524: Gompers, P Optimal investment, monitoring, and the staging of venture capital. The journal of finance, 505: Guesnerie, R., Picard, P., and Rey, P Adverse selection and moral hazard with risk neutral agents. European Economic Review, 334: Hart, O. and Moore, J Default and renegotiation: A dynamic model of debt. The Quarterly Journal of Economics, 1131:1 41. Holmstrom, B. and Tirole, J Financial intermediation, loanable funds, and the real sector. The Quarterly Journal of Economics, 1123: Holmstrom, B. and Tirole, J Private and public supply of liquidity. The Journal of Political Economy, 1061:1 40. Kashyap, A., Stein, J., and Wilcox, D Monetary policy and credit conditions: Evidence from the composition of external finance. The American Economic Review, pages Mailath, G. J. and Samuelson, L Repeated games and reputations: Long-run relationships. Oxford University Press. Repullo, R. and Suarez, J Venture capital finance: A security design approach. Review of Finance, 81: Sahlman, W The structure and governance of venture-capital organizations. Journal of Financial Economics, 272: Townsend, R Optimal contracts and competitive markets with costly state verification. Journal of Economic Theory, 212: Williamson, S Costly monitoring, loan contracts, and equilibrium credit rationing. The Quarterly Journal of Economics, pages A.1. Proof of proposition 3 A. Proofs Let σ = α, µ 1, µ 2 be a PPE and fix a history ht with α 1t ht = 1. Define m i ξ = µ it h t, ξ, u ξ = u 1 ξ, m 1 ξ, m 2 ξ as in equation 2, V = V 1 σ h t as 1 s expected discounted payoff and V ξ = V 1 σ h t,ξ,mξ as the continuation values. So that: V = 1 δd + π 1 δv δu π 1 δv δu For α h t to be optimal, choosing a = 0 must be no better than a = 1 which implies: δv δu 1 δv δu 0 1 δ ˆd 19 15

16 After ξ = 1 is realized, player 1 could guarantee himself a cost of not more than c 1 this period and his minimax payoff π 0 c 1 from t + 1 onwards: δv δu 1 1 δ π 0 δc 1 20 Combining both incentive constraints we have that: δπ 0 ˆd δv δu 0 1 δ π 0 δ c 1 ˆd 21 Substituting 18 in 19 and doing some algebra: δπ 0 ˆd δ 2 1 δ V 0 Combining 21 and 22 and doing some more algebra we obtain: δ V + δu δ V 0 1 δ V + γ, γ 1 δ π0 δ 1 δ ˆd c1 δ 23 Hence we have established that whenever the active agent chooses a = 1 and there is no crisis, his expected value must increase at least by a fixed factor γ. The assumption ˆd > c 1 guarantees that γ > 0. This implies that if there is no crisis for n subsequent periods and the active agent keeps choosing a = 1 with probability 1 we must have V n > V + nγ. 24 Where V n is 1 s expected discounted value at period t + n. Since the set of feasible payoffs of agent 1 is bounded above by 0, it must be the case that after a long enough history of no crisis agent 1 takes non avoidance action. Otherwise V n would be greater than 0. On the other hand, any finite sequence of no crisis occurs with positive probability. Therefore, taking the avoidance action at every period with probability 1 cannot be part of a PPE. A.2. Proof of proposition 4 Consider a strategy profile such that for every history h t, either µ 1t h t, ξ t c 1 or µ 1t h t, ξ t + µ 2t h t, ξ t < 1. So that v 1 σ t h t d π 1 c 1 if α t h t = 1 and v 1 σ t h t π 1 c 0 if α t h t = 0. Assumption 2 implies that d π 1 c 1 < π 0 c 1, and thus V 1 σ π 0 c 1, with strict inequality whenever α t h t = 1 for some history h t which is reached with positive probability along the equilibrium path. Individual rationality requires V 1 σ π 0 c 1, and thus it can be satisfied if and only if α t h t = 0 almost surely along the equilibrium path. 16

17 A.3. Proof of proposition 5 Let σ be a PPE such that µ 1th t, 1 + µ 2th t, 1 < 1 for some history h t that is reached with positive probability. Consider the alternative strategy profile σ with µ 1t ht, 1 = µ 1t ht, 1 + c 1 and µ 2t ht, 1 = max{0, 1 µ 1t ht, 1}. We claim that σ is also a PPE and Pareto dominates σ. Notice that the cost for player 1 according to σ and σ coincide in every period after any history. The cost for 2 is also the same for every period and history other than h t. If there is a crisis in period t and given h t, the cost for 2 corresponding to σ is µ 2t ht, 1 c 2 and according to σ is µ 2th t, 1 µ 1th t, 1 1 µ 1th t, 1 1 < µ 2th t, 1 < c 2 µ 2th t, 1. This implies that σ Pareto dominates σ, and that, in order to verify that σ is an equilibrium, we only need to check that there are no profitable deviations after h t, 1. For 2 there are no profitable deviations because µ 2t ht, 1 is a static best response to µ 1t ht, 1. For 1 there are two possibilities. If µ 2t ht, 1 > 0 then µ 1t ht, 1 is already a static best response to µ 2th t, 1. If µ 2th t, 1 = 0, then the best possible deviations where also available given σ. Since σ is an equilibrium, then so is σ. A.4. Proof of proposition 6 We start with some notation and preliminaries. Throughout the proof we only use strategy profiles with efficient mitigation, i.e. strategy profiles such that µ 1t h t, 0 = µ 2t h t, 0 = 0 and µ 1t h t, 1 + µ 2t h t, 1 = 1. 6 A strategy profile for the pure game without money burning can be described by a tuple a, m 1, m 2 where m i represents i s contribution in the event of a crisis. We use the notation a, m 1, m 2 to denote the strategy of the repeated game which consists of repeating a, m 1, m 2 after any arbitrary history. Notice that 0, m 1, 1 m 1 is a PPE as long as m 1 [1 c 2, c 1 ]. Given a strategy profile σ = α, µ 1, µ 2, in order to analyze the incentive constraints at period t given h t, we use the notation m i = µ it h t, 1 to denote the mitigation contributions if there is a crisis at period t, and V ξ i = V i σ h t,ξ for the continuation values. Also, we only consider strategies which go to the minimax equilibrium 0, 0, 0 after any observable deviation. With this in mind, and with mitigation and no money burning, the incentive constraints for mitigation if m i > 0 can be written as: 1 δm i + δv 1 i 1 δ + δπ 0 c i 25 If a = 1, then the incentive constraint for avoidance can be written as: δv 0 1 V δ ˆd m We don t change the game as to preclude the possibility of burning money or no-mitigation. We simply don t use such strategies in our constructions. 17

18 We make use of the strategy profile σ 0 described as follows: σ1 0 = 1, c 1, 1 c 1 It there is a crisis in period one, then σt 0 = 0, c 1, 1 c 1 for t > 1. It there is no crisis in period one, then σt 0 = 0, 1 c 2, c 2 for t > 1. Also, we make use of the following two bounds for the discount factor: δ ˆd c 1 ˆd c 1 + π 0 c 1 + c 2 1, δ ˆd c1 ˆd c 1 + π 0 c 1 27 The assumptions that ˆd > c 1, c 1 + c 2 > 1, and c 2 < 1 imply that 0 < δ < δ < 1. Lemma 7 σ 0 is a PPE if and only if δ δ. Proof. For t > 1, σ 0 is just repetition of static Nash equilibria and hence the continuation strategies are PPE. If there is a crisis in period 1, the agent s contributions are static best responses and do not affect the continuation value, hence the mitigation incentive constraints are satisfied. We only need to check the avoidance incentives for agent 1 in period 1. We have V1 0 = π 0 1 c 2 and V1 1 = π 0 c 1. And hence V1 0 V1 1 = π 0 c 1 + c 2 1. Therefore the avoidance constraint 26 can be written as: δv 0 1 V δ ˆd c 1 δπ 0 c 1 + c 2 1 ˆd c 1 δ ˆd c 1 δ [ π 0 c 1 + c ˆd c 1 ] ˆd c 1 δ δ Given these preliminaries, we can proceed to the main body of the proof, which is organized as follows. Lemma 8 states that whenever it is possible to play avoidance at least once in at least one PPE, then it can be done in an efficient manner which reduces the total cost. Lemma 9 then states that in that case, avoidance can improve the payoff of player 2 without hanging the payoff of player 1. As a consequence, whenever it is possible to play avoidance, Pareto efficiency requires doing so infinitely often. Finally, we show that if δ δ it is not possible to play avoidance, and Lemma 7 implies that if δ δ it is possible to play avoidance. Hence the avoidance threshold δ 0 lies within 0, 1. Lemma 8 If it is possible for agent 1 to choose the avoidance action with positive probability at least once in at least one PPE, then there exists a PPE with total cost less than π 0, i.e. there exists V 1, V 2 V such that V 1 + V 2 > π 0. Proof. Suppose that σ is a PPE and after some history we have α t h t = 1. There are two possibilities, if the continuation values are such that either V1 0 + V 2 0 > π0 or V1 1 + V 2 1 > π 0 then the continuation strategies satisfy the conclusion of the lemma and we are done. 18

19 Otherwise, we must have V1 0 + V2 0 π 0 and V1 1 + V2 1 π 0. This implies that V 1 and V 0 can be achieved by some strategy profiles 0, m 0 1, m0 2 and 0, m1 1, m1 2 respectively, i.e. we must have V ξ i = π 0 m ξ i for i = 1, 2 and ξ = 0, 1. Also, since we are trying to minimize total cost and increasing the payoff of agent 2 only relaxes the incentive constraints, we can assume without loss of generality that m 2 = 1 m 1, and m ξ 2 = 1 m ξ 1. Substituting in 25 and 26, the incentive constraints for agent 1 at period t given h t yield: 1 δm 1 + δπ 0 m δ ˆd + δπ 0 m δm 1 + δπ 0 m δ + δπ0 c 1 1 δ ˆd + δπ 0 m δ + δπ0 c 1 Individual rationality requires that V 0 2 = π0 1 m 0 1 π0 c 2, which in turn implies that m c 2 and thus: 1 δ ˆd + δπ 0 1 c 2 1 δ + δπ 0 c 1 After some simple algebra, this implies that δ δ and therefore, by Lemma 7, σ 0 is a PPE and V 1 σ 0, V 2 σ 0 V. Finally, notice that V 1 σ 0 +V 2 σ 0 = 1 δd+π 1 δπ 0 > π 0, where the inequality follows because assumption 2 implies that d + π 1 < π 0. Lemma 9 If there exists a PPE with total cost less than π 0, then every PPE without avoidance is Pareto dominated by a different PPE which increases the payoff of agent 2 while keeping the payoff of agent 1 unchanged, i.e. for every PPE σ such that α t h t = 0 almost surely, there exist V 1, V 2 V such that V 1 = V 1 σ and V 2 > V 2 σ. Proof. We will show that there exist two PPE σ 1 and σ 2 such that V 1 σ 1 = π 0 c 1 and V 2 σ 1 > π 0 1 c 1, and V 1 σ 2 > π 0 1 c 2 and V 2 σ 2 = π 0 c 2. Now consider any m 1 [1 c 2, c 1 ] and the strategy profile σ = 0, m 1, 1 m 1. The corresponding payoff vector is V i σ = π 0 m 1, π 0 1 m 1. Since 1 c 2 m 1 c 1, it follows that V 1 σ 1 = π 0 c 1 π 0 m 1 π 0 1 c 2 < V 1 σ 2, and hence we can write V 1 σ = λv 1 σ λv 1 σ 2 with λ 0, 1]. Now consider the strategy profile σ that plays σ 1 with probability λ and σ 2 with probability 1 λ. By construction we have that V 1 σ = V 1 σ. Also we know that V 1 σ + V 2 σ > π 0 = V 1 σ + V 2 σ, and therefore V 2 σ > V 2 σ. Hence it only remains to show the existence of σ 1 and σ 2. Let σ be a PPE with total expected discounted cost less than π 0, i.e. such that V 1 σ + V 2 σ > π 0. Individual rationality implies that V i σ 0 π 0 c i for i = 1, 2, and we assume that both the inequalities are strict the proof can be easily adapted to accommodate the general case. Hence we can write V 1 σ = π 0 m 1 1 and V 2 σ = π 0 1 m 1 2, for some m 1 [1 c 2, c 1 ] and 1, 2 > 0. Also, notice that by mixing σ with 0, m 1, 1 m 1 we obtain a new PPE with payoffs between those of σ and 0, m 1, 1 m 1. Hence we can assume without loss of generality that 1, 2 are arbitrarily small. 19

20 For ε 0, 1 c 1, consider the strategy profile σ ε defined as follows: σ ε 1 = 0, c 1 + ε, 1 c 1 ε It ξ 1 = 1 then σt ε = σ1 t 1 for all t > 1 It ξ 1 = 0 then σ ε t = 0, 1 c 2, c 2 for all t > 1 The continuation strategies after period 1 are PPE. In period 1, agent 2 plays a static best response, and his contributions do not affect the continuation value. Hence we only need to check the incentives for agent 1 in period 1. The corresponding mitigation constraint 25 can be written as: 1 δc 1 + ε δπ 0 m δc 1 δπ 0 c 1 δ ε ε π c 1 m 1 > 0 1 δ The corresponding avoidance constraint for α 1 = 0 can be written as the opposite of 26: δv 0 1 V 1 1 = δπ 0 m δπ 0 1 c 2 1 δ ˆd m 1 28 Using the fact that m 1 [1 c 2, c 1 ], it is straightforward to verify that: δv 0 1 V 1 1 δπ δ ˆd m 1 1 δ ˆd c 1 Hence, a sufficient condition in order to satisfy the avoidance constraint 28 is that: δπ δ ˆd c 1 D 1 1 δ ˆd c 1 δπ 0 > 0 Which can be satisfied because we have already established that we can assume without loss of generality that 1 is small enough. Hence σ 1 = σ ε is a PPE. The corresponding payoff for player 1 is: V 2 σ 1 =π 0[ 1 δc 1 + ε δπ 0 m 1 1 ] + 1 π 0 [ δπ 0 c 1 ] = π 0 c 1 Finally, since the total cost of σ is less than π 0, it follows that so is the total cost of σ 1 and therefore: V 2 σ 1 > π 0 V 1 σ = π 0 + π 0 c 1 = π 0 1 c 1 Hence σ 1 is a PPE satisfying the desired properties. The proof for σ 2 is almost analogous. The only difference is that the relevant avoidance constraint in the construction can also be satisfied when δ is low enough. This is not 20

21 a problem because it can be satisfied whenever δ δ and in particular when δ = δ the conditions of the Lemma are also satisfied by lemmas 7 and 8. Since the set of equilibrium payoffs is monotone with respect to δ, this implies that whenever δ > δ there will be some PPE which achieves the same payoff vector. Now we are finally in position to prove Proposition 6 proof of Proposition 6. i Suppose that σ is a PPE and after some history the active agent chooses α t h t = 1. If µ 1t h t > 0 then the avoidance and mitigation constraints require: 1 δm 1 δv δ + δπ0 c 1 1 δm 1 δv1 1 1 δ ˆd δv1 0 1 δ ˆd δv δ + δπ 0 c 1 If µ 1t h t = 0 then there is no mitigation constraint, but individual rationality requires δv1 1 δπ 0 c 1 1 δ + δπ 0 c 1 and we obtain the same inequality. Furthermore, feasibility implies that V After some algebra this implies that δ δ 0, 1. Hence if δ is low enough there is no avoidance. ii Let 0, 1 be the set of discount factors for which it is possible to sustain avoidance. Lemma 7 implies that. Lemma 8 implies that for any δ, there exits a PPE with total cost below π 0. Since the set of PPE payoffs is monotone with respect to the discount factor, the same is true for every δ δ. Since obtaining a cost below π 0 requires avoidance, this implies that is an interval. Let δ 0 = inf, we have established that δ 0, 1. I am not sure whether δ 0, but I don t think that this is relevant. Part i and Lemma 7 imply that 0 < δ δ 0 δ < 1. For the remainder of the proof fix any δ > δ 0 and let σ be a constrained Pareto efficient PPE given δ. Suppose towards a contradiction that there exists a set of histories which is reached with positive probability after which there is no avoidance. For every history in teh set let σ be the continuation strategies. By Lemma 8 we know that the consequence of Lemma 9 apply, and hence there exists some PPE σ such that V 1 σ = V 1 σ and V 2 σ > V 2 σ. If we replace σ with σ in every such history, all the incentive constraints of previous periods remain unchanged. Hence we obtain a PPE which Pareto dominates σ. Therefore we can conclude that if σ is constrained Pareto Efficient, there there is avoidance infinitely often almost surely. By Proposition 4, this implies that there are also bailouts infinitely often almost surely. B. APS characterization Lemma 10 [Single deviation principle] An individually rational strategy profile is a PPE if and only if it admits no profitable single-deviations. 21

22 Proof. The result follows can be proven in a similar way as Proposition in Mailath and Samuelson 2006, pp 25. The general lines of the argument are as follows. Fix an individually rational strategy profile and suppose that there is a profitable deviation and let V be the difference in values. Since the set of individually rational payoffs is bounded, we know that there is some T such that the payoffs after T contribute to less than V /2 and thus there is also a profitable deviation of length at most T. If the deviation in the last period is profitable, then the proof is complete. If not, then there is a profitable deviation of length at most T 1 periods. By induction, this implies that there is a profitable deviation of length 1. Lemma 11 [Self-generation] If W is bounded and self-generating then W V Proof. For each v W, by Carateheodory s theorem, there exist some b 1 v, b2 v, b3 v sb s W and λ 1 v, λ 2 v, λ 3 v 3 such that v = 3 n=1 λ n vb n v. For each b n v, we can define s n v and wv n so that b n v = gsn v, wn v and sn v, wn v is admissible w.r.t. W. Now fix some v W. We will construct a PPE σ such that v = V σ. For that purpose we define a sequence of public history-dependant continuation values {wt } with wt : H t W. σ is defined as a function of w : s 1 wt ht if θ t λ 1 wt ht σt ht, θ t = s 2 wt ht if λ 1 wt ht < θ t λ 2 wt ht s 3 wt ht if θ t > λ 2 wt ht w is defined recursively with w 1 = v and: w w w 1 t ht m t, ξ t if θ 1 λ 1 wt ht t+1 h t+1 = ww 2 t ht m t, ξ t if λ 1 wt ht < θ 1 λ 2 wt ht ww 3 t ht m t, ξ t if θ 1 > λ 2 wt ht It is straightforward to see that w T and thus σ t are measurable. Now notice that for every h t : 3 wt h t = λ n wt g s n ht wt, w n ht wt ht = Et [g σt h t, θ t, wt+1h t+1 ] n=1 [ = Et 1 δv σt ht, θ t ] + δwt+1 ht+1 [ = 1 δet vσt + δvσ t+1 + δ ] 1 δ w t+2 ht+2 =... = V σ t t And hence we have that v = w 1 = V σ. Finally, since actions and continuation values 22

23 are admissible at every period, we know that there are no profitable single-deviations. By Lemma 10 this implies that σ is a PPE. Lemma 12 [Factorization] V is self-generating Proof. Fix an arbitrary point v V and let σ be the PPE that generates it. For each θ [0, 1], m 1, m 2 M and ξ X set w θ m 1, m 2, ξ = V σ θ,ξ,m1 ξ,m 2 ξ V. Since σ is measurable it follows that w θ is measurable for θ [0, 1]. Since σ is a PPE we know that for each realization of θ 1 we have that σ1 θ 1, w θ1 is admissible w.r.t. V and thus g σ1θ 1, w θ1 B σ1 θ 1 V. This implies that: 1 v = g σ1 θ 1, w θ1 dθ 1 co B s V = BV 0 s Lemma 13 If W is compact then BW is compact Proof. Fix some a A. We will start by showing that B a W m B a,m W is compact. Consider any sequence {v n } in B a W converging to some v R 2. By construction there exist sequences {m n } and {w n } such that v n = ga, m n, w n and a, m n, w n is admissible w.r.t. W. Since it is contained in a compact space, the sequence {m n, w n m n } has a subsequence converging to some limit m, w m. Since W and M are closed, we know that m : X M and w m W. Let wi m, m i ξ, ξ = min w W w i for m m i ξ and define other values of w arbitrarily. Since g is continuous we know that v = ga, m, w. Since the incentive constraints are defined by continuous functions we know that and a, m, w is admissible w.r.t. W. Hence v B a W. Since this was for arbitrary convergent sequences, this means that B a W is closed. Now, since the payoffs of the stage game are all non-positive and W is bounded, then B a W is bounded above. Since admissibility implies that the values have to be conditionally individually rational, it is also bounded below. Hence B a W is compact. Since a finite union of compact sets is compact, we have that s B s W = a B a W is compact. The result then follows from the fact that the convex hulls of compact sets are compact. Proof of proposition 1. Since B is -monotone by construction, Lemma 11 implies that V contains the union of all self-generating sets. By Lemma 12 this implies that V is the largest self-generating set. Since BW is convex for any W by construction, Lemma 12 also implies that V is convex. Now fix any bounded set W such that V W. Let W be the closure of W and define the sequence {W n } n=1 by W 0 = W and W n+1 = BW n for n = 1, 2,.... By definition of B and Lemma 13 we know that W n is a -decreasing sequence of compact 23