Contagious Adverse Selection

Transcription

1 Stephen Morris and Hyun Song Shin Princeton Economics Department Seminar November 2008

2 Credit Crunch "market condence"

3 Credit Crunch "market condence" undermined by unexpected losses (in housing and some related nancial instruments)

4 Credit Crunch "market condence" undermined by unexpected losses (in housing and some related nancial instruments) "opaqueness" of those nancial instruments

5 Credit Crunch "market condence" undermined by unexpected losses "opaqueness" of those nancial instruments an interpretation: "opaqueness" no accident: informational rents for nancial intermediaries informational rents imply uninformed traders faced adverse selection (broadly interpreted)

6 Credit Crunch "market condence" undermined by unexpected losses "opaqueness" an interpretation: "opaqueness" no accident: informational rents for nancial intermediaries informational rents imply uninformed traders faced adverse selection but markets operated OK with adverse selection in good times

7 A Propagation Channel Shock to system shows that common understanding of background adverse selection is wrong in asset market A A few (pessimistic) uninformed traders drop out of market A A few other traders - thinking that uninformed traders on the other side of the market are dropping out - also drop out and so on A few (pessimistic) uninformed traders drop out of market B, which they think may be correlated with market A and so on...

8 Contagious adverse selection propagation channel: Shock to system shows that common understanding of background adverse selection is wrong in asset market A A few (pessimistic) uninformed traders drop out of market A A few other traders - thinking that uninformed traders on the other side of the market are drop out - also drop out and so on A few (pessimistic) uninformed traders drop out of market B, which they think may be correlated with market A and so on... c.f., other propagation channels, e.g., Kiyotaki and Moore (1998) understand a contagious adverse selection channel in a clean simple theoretical model

9 Higher Order Adverse Selection Uninformed are unsure if others in market are "informed" "Adverse selection" is selection of informed market partners n Akerlof (1970), all sellers are informed: "Adverse Selection" is selection of informed sellers with bad cars. Leads to coordination problem between uninformed agents (uninformed agents want to trade only if other uninformed agents trade)

10 Contagious Miscoordination (driven by Adverse Selection) Coordination on risky outcomes possible if and only if approximate common knowledge of gains from coordination Lack of common knowledge of gains from coordination implies contagious spreading of inecient outcomes Thus market condence = approximate common knowledge of upper bound on expected losses of uninformed agents

11 Akerlof (1970) meets Rubinstein (1989) George Akerlof, QJE 1970, "The Market for 'Lemons'" Ariel Rubinstein, AER 1989, "The Electronic Mail Game"

12 Accounting Standards / Credit Ratings One purpose is to provide sucient common understanding of "value" of underlying asset to allow impersonal transactions (Morris and Shin 2007 "Optimal Communication") Or create "market condence" i.e., approximate common knowledge of upper bound on expected losses This was our original motivation for understanding contagious adverse selection: identifying model-based social value of widely available and focal information sources

13 This Talk Present "minimal" model of higher order adverse selection leading to contagious miscoordination (work in progress) Review results on common knowledge and coordination and why we think they are of applied interest (research agenda review) Discussing accounting agenda and implications (if any) for credit crunch

14 Ecient Bilateral Trade Agent 1's private value of a house is v c Agent 2's private value of a house is v + c Agent 1 owns the house Ecient for 1 to sell the house to 2, say at price v

15 Adding Adverse Selection But the house may be falling down, in which case the seller (agent 1) knows it and the values are (v M c, v M + c) respectively Or the buyer (agent 2) may be developer may be planning a new complex, in which case he knows it and the values are (v + M + c, v + M c) respectively Potential losses bigger than certain gains M c

16 Adding Adverse Selection So total values are value to agent 1 value to agent 2 "normal" state v c v + c 1's informed state v M c v M + c 2's informed state v + M c v + M + c

17 Adding Adverse Selection So total values are value to agent 1 value to agent 2 "normal" state v c v + c 1's informed state v M c v M + c 2's informed state v + M c v + M + c f agents trade at price v, total gains from trade are gains of agent 1 gains of agent 2 "normal" state c c 1's informed state c + M c M 2's informed state c M c + M

18 Adding Adverse Selection Uninformed agent assigns probability 1 probability δ to partner's informed state δ to normal state, gains of 1 gains of 2 prior prob. 1 δ "normal" state c c 1+δ δ 1's informed state c + M c M 1+δ δ 2's informed state c M c + M 1+δ

19 The Loss Ratio "Loss Ratio" of uninformed agents: ψ = expected losses gains = δm c renormalization: x gains from trade (c) and loss ratio (ψ), vary absolute losses (M) and let the probability of losses be δ = cψ M later introduce uncertainty about ψ

20 Welfare Ex Ante Welfare = 2c Probability of Trade

21 Complete nformation Complete nformation about loss ratio ψ There is still background asymmetric information about whether we are in normal state or not

22 Trading Game each agent says "yes" or "no" if and only if both say "yes," trade takes place informed types always trade (dominant strategy)

23 Trading Game each agent says "yes" or "no" if and only if both say "yes," trade takes place informed types always trade uninformed types play coordination game if uninformed partner trades, expected payo to trade is cψ 1 c + cψ (c M) M M cψ = c M = c (1 ψ) M if uninformed partner does not trade, expected payo to trade is cψ M = cψ (c M) c 1 M < 0

24 Simple Trading Game Yes No c Yes c (1 ψ), c (1 ψ) cψ 1 M, 0 c No 0, cψ 1 M 0 f loss ratio ψ 1, (Yes, Yes) is an equilibrium with probability of trade 1 f loss ratio ψ > 1, (No, No) is a unique equilibrium with probability of trade 0

25 Simple Trading Game as M! Yes No Yes c (1 ψ), c (1 ψ) cψ, 0 No 0, cψ 0 f loss ratio ψ 1, (Yes, Yes) is an equilibrium with probability of trade 1 f loss ratio ψ > 1, (No, No) is a unique equilibrium with probability of trade 0

26 Optimal Mechanism Suppose loss ratio ψ > 1 (so no trade in simple trading game) Trade with probability 1 in informed state, with probability q 2 (0, 1) in normal state nformed agent pays p to uninformed agent in his informed state Binding individual rationality constraint of uninformed agent: cψ 1 c + cψ (c M) + p = 0 M M Binding incentive compatibility of informed agent: c + M p = q (c + M)

27 Optimal Mechanism Solving gives: p = (1 q) (c + M) 2c q = 2c + M Total probability of trade is 2cψ M + 2cψ 1 M 1 1 ψ 2c 2c ψ M Bottom line: as M!, probability of trade decreases to 0.

28 ncomplete nformation There is background asymmetric information about whether we are in normal state or not ALSO there is a lack of common knowledge of loss ratio ψ

29 Example: Uncertainty Uncertainty about loss ratio Ω = f1, 2,..., 2K + 1g, uniform prior Loss ratios ψ 1... ψ 2K +1 NB: for each ω state, three substates: normal, 1's informed, 2's informed Almost perfect information: Agent 1 observes partition (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) Agent 2 observes partition (f1, 2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g)

30 Example: Losses Losses very high in state 1 Losses quite high but less than 1 in other states ψ 1 = ψ H 1 ψ ω = ψ L 2 1 2, 1 for ω 2

31 Example: Summary Ω = f1, 2,..., 2K + 1g, uniform prior ψ 1 = ψ H 1; ψ ω = ψ L 2 1 2, 1 for ω 2 Almost perfect information: Agent 1 observes partition (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) Agent 2 observes partition (f1, 2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g)

32 Simple Trading Game Uninformed type f1g of agent 1 does not trade in equilibrium Uninformed type f1, 2g of agent 2 does not trade in equilibrium Candidate equilibrium: agent 1 trades if ω 2 agent 2 trades if ω 3

33 Simple Trading Game Expected payo of uninformed type f2, 3g of agent 1 in proposed equilibrium: 1 cψl 1 cψ 2 M 2 1 L 1 cψl M 2 M 1 + cψ (c M) + L 1 M cψ c + (c M) L 1 M cψ L M 1 2 = c 1 2 cψ L M 1 + cψ L M M = c! c (1 2ψ L ) as M! < 0 2cψ L M M + cψ L Unique equilibrium has no trade (Rubinstein (1989))

34 General nformation Structure Finite state space Ω example: Ω = f1, 2,.., 2K + 1g Prior π 2 (Ω) example: π (ω) = 1 2K +1 Loss ratio ψ : Ω! R + for all ω example: ψ (1) = ψ H and ψ (ω) = ψ L for ω 2 Partitions P 1, P 2 example, e.g., P 1 = (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) Write P i (ω) for set of states i thinks possible example, e.g., P 1 (3) = f2, 3g

35 Higher Order Beliefs Write eψ i (ω) for agent i expected loss ratio at state ω: eψ i (ω) = π (ω 0 ) ψ (ω 0 ) ω 0 2P i (ω) π (ω 0 ) ω 0 2P i (ω) in example: eψ 1 (1) = ψ H and eψ 1 (ω) = ψ L 8 ω 2 eψ 2 (1) = eψ 2 (2) = 1 2 (ψ H + ψ L ) and eψ 2 (ω) = ψ L 8 ω 3

36 Higher Order Beliefs Fix event E Ω. Write B ψ i (E ) for states ω where i believes that event E occurs with probability at least eψ i (ω): B ψ i (E ) = ω π (E jp i (ω) ) eψ i (ω) π (ω 0 ) ω where π (E jp i (ω) ) = 0 2E \P i (ω) π (ω 0 ) ω 0 2P i (ω) example: B ψ 2 (f2, 3,..., 2K + 1g) = f3,..., 2K + 1g

37 Higher Order Beliefs Write B ψ (E ) for states ω where each agent believes that event E occurs with probability at least eψ i (ω): example: B ψ (E ) = B ψ 1 (E ) \ B ψ 2 (E ) B ψ (f2, 3,..., 2K + 1g) = f3,..., 2K + 1g B ψ (fk, k + 1,..., 2K + 1g) = fk + 1,..., 2K + 1g for each k = 2, 3,.. B ψ (f2k + 1g) =?

38 Market Condence There is "market condence" at ω if 1. each agent's expected loss ratio is less than 1 2. each agent i believes (1) with probability at least eψ i (ω) 3. each agent i believes (2) with probability at least eψ i (ω) write C ψ for the set of states where there is market condence h i C ψ = B ψ (Ω) \ B ψ 2 (Ω) \... h i = \ B ψ k (Ω) k1 Say that event E is ψ-evident if E B ψ (E ). PROPOSTON. C ψ is the largest ψ-evident event. Aumann (1976), Monderer-Samet (1989), Morris-Shin (2007).

39 Market Condence in Example h B ψ h h B ψ B ψ Ω = f1, 2, 3,..., 2K + 1g B ψ (Ω) = f2, 3,..., 2K + 1g i 2 (Ω) = f3, 4,..., 2K + 1g i n (Ω) = fn + 1,..., 2K + 1g i 2K +1 (Ω) =? C ψ =? There is never market condence in the example.

40 Simple Trading Game PROPOSTON. 1. There is always a no trade equilibrium. 2. There is a "largest" trade equilibrium. 3. n the largest equilibrium, informed agents always trade. 4. Uninformed agents trade only if there is market condence. 5. For suciently large M, they trade if and only if there is market condence.

41 Optimal Mechanism (with small transaction costs) Fix an event E and write E i = fω jp i (ω) \ E 6=?g. Event E is ψ-evident if π (E jp i (ω) ) eψ i (ω) for all i and ω 2 E i Event E is weakly ψ-evident (Geanakoplos 1994) if π (E 2 je 1 ) + π (E 1 je 2 ) Exp (ψje 1 ) + Exp (ψje 2 ). Write W ψ for the largest weakly ψ-evident event There is weak market condence at ω if ω 2 W ψ. PROPOSTON. f W ψ =?, then for each ε > 0, there exists M > 0 such that the ex ante probability of trade in the optimal mechanism is at most ε.

42 Bound in Example Ω = f1, 2,.., 2K + 1g, uniform prior ψ 1 = ψ H 1; ψ ω = ψ L 2 1 2, 1 for ω 2 Almost perfect information: Agent 1 observes partition (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) Agent 2 observes partition (f1, 2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g) small trade for large M if ψ L > 1 1 2K

43 Bound in Example K = 1 Ω = f1, 2, 3g, uniform prior ψ 1 = ψ H 1; ψ 2 = ψ 3 = ψ L Almost perfect information: Agent 1 observes partition (f1g, f2, 3g) Agent 2 observes partition (f1, 2g, f3g) small trade for large M if ψ L > 3 4

44 dea of Proof in Optimal Mechanism Bound Bound probability of trade using only ndividual Rationality Constraints of Uninformed Types ncentive Compatibility Constraints of nformed Types Assume w.l.o.g. trade with probability 1 in informed states, q (ω) in normal state ω Agent i in informed state gets rent M 1 max ω 0 2P i (ω) q (ω 0 ) High q implies rents bigger than 2c, contradiction

45 dea of Proof in Optimal Mechanism Bound Bound probability of trade using only ndividual Rationality Constraints of Uninformed Types ncentive Compatibility Constraints of nformed Types Assume w.l.o.g. trade with probability 1 in informed states, q (ω) in normal state ω Agent i in informed state gets rent M 1 max ω 0 2P i (ω) q (ω 0 ) High q implies rents bigger than 2c, contradiction NOTE: despite correlated types, no full surplus extraction (Cremer-McLean 88) because linearly dependent beliefs (Neeman 04, Parreiras 05)

46 Future Work on Optimal Mechanism tightness? useful mechanisms that do better than simple trading mechanism? richer higher order beliefs limit returns to more complex mechanisms?

47 Recognition versus Disclosure Recognition: in nancial bottom line Disclosure: in the footnote Makes a dierence [Barth, Clinch and Shibano (2003), Espahbodi et al. (2002)]

48 Accounting Standards Finding the balance between accurate measurement common understanding Attempting to convey more information may lead to less focussed attention and less common understanding One purpose of accounting standards is to signal to audience what numbers others will simultaneously be focussing on

49 Back to the Example Ω = f1, 2,..., 2K + 1g, uniform prior ψ 1 = ψ H 1; ψ ω = ψ L 2 1 2, 1 for ω 2 what is the social value of dierent information structures concerning losses? choose P = (P 1, P 2 ) to maximize C ψ P subject to P 2 F

50 Low loss ratio in bad state ψ H not too high no information P 1 = P 2 = (f1, 2, 3,..., 2K + 1g) is optimal information structure

51 High loss ratio in bad state ψ H is very high, unconstrained optimal information structure is P 1 = P 2 = (f1g, f2, 3,..., 2K + 1g) with noisy interpretation constraint that information must coarsen P 1 = (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) P 2 = (f1, 2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g) optimal information structure is P 1 = (f1g, f2, 3,..., 2K + 1g) P 2 = (f1, 2g, f3, 4,..., 2K + 1g)

52 High loss ratio in bad state ψ H is very high, with background information constraint that information must rene P 1 = (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) P 2 = (f1, 2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g) optimal information structure is P 1 = (f1g, f2, 3g,..., f2k 2, 2K 1g, f2k, 2K + 1g) P 2 = (f1g, f2g, f3, 4g,..., f2k 1, 2K g, f2k + 1g)

53 Credit Crunch (Outside of Model) Observations Original TARP concept - "getting toxic assets o the balance sheet" - not so crazy? Credit cycles resulting from endogenous adverse selection: market participants have incentive to pool as much adverse selection as market will bear, bringing system to critical point? "Common rank beliefs" = lack of common knowledge giving rise to contagion in coordination games (Morris and Shin 2007). f public shocks give rise to common rank beliefs, there is strategic propagation mechanism of shocks

54 Conclusion of Work in Progress Analyzed a particular two sided adverse selection model Distinctive ingredient: unravelling along dierent assessments of expected losses New Directions Accounting / Credit Rating Application Mechanism Design Results Develop insights within standard nancial models