Experiments with Network Economies

Dean Corbae University of Texas John Duffy University of Pittsburgh

Questions

• How do agents choose their trading networks? • Why do certain trading strategies spread? — The answers have important implications for propagation of shocks.

• Why use experiments? — Hard to get good micro data on network formation and contagion.

Motivation • Allen & Gale (2000 JPE) “Financial Contagion”. — Network version of Diamond-Dybvig (1983 JPE) banking model (strategic complementarities). — 4 regions composed of ex-ante identical agents who receive unobservable preference shocks. — Fractions of (im)patient agents vary across regions in 2 different states (no agg. uncertainty). — Regions linked by different networks of interbank deposit markets. ∗ Complete (uniform matching) ∗ Incomplete (local interaction or marriage). — Agg. excess liquidity demand shock causes a contagious bank run in LI network.

Literature

• Theory — Exogenous Networks: Ellison (1993, Ecta), Kandoori, Mailath, Rob (1993, Ecta), (Morris (2000, REStud), Young (1993, Ecta). — Endogenous Networks: Bala and Goyal (2000, Ecta), Jackson and Watts (2002, JET), Jackson and Wolinksy (1996, JET).

• Experiments — Exogenous Local Interaction in a Minimum Effort game: Keser, Ehrhart, Berninghaus (1998, EL), Berninghaus, Ehrhart, Keser (2002, GEB). — Deviations from Marriages: Hauk and Nagel (2002).

Outline of Talk

1. Environment

2. DeÞne Equilibrium (Perfect Bayesian)

3. Predictions

4. Experimental Evidence

• Exogenous Networks (Study Contagion) — Morris (2000) deÞnes contagion as “the spread of one action from a Þnite set of players to the whole population.”

• Endogenous Networks

Environment (SimpliÞed)

• Population: 4 agents I = {1, 2, 3, 4}. • Timing: A Þnite sequence of two-stage games. — First (Proposal) stage: Link proposals are made and a network is formed. — Second (Investment) stage: τ rounds of investment decisions which are payoff relevant.

• Network Matching. — Each agent sends link proposals to all other agents pi = (pij )4j=1 ∈ {0, 1}4. — Mutually agreed upon links are formed (i.e. j link ij occurs iff pij pi = 1). — A network is just the set of all agreed upon links (g = {ij : ∀i, j ∈ I} ∈ Γ, where Γ is the set of all possible networks). — For simpliÞed version, network formation is costless.

— Agent i’s neighborhood is the set of all agents to whom he is linked (N i(g) = {j|ij ∈ g, j 6= i}). The number of neighbors of agent i is simply the cardinality of N i(g) (denoted ni(g)). — Matching weights given by: µij (g) =

  

1

ni (g)

0

if j ∈ N i(g), otherwise.

(1)

• Investment Actions. — For τ rounds, each agent makes “risky” (R) or “safe” (S) investment choices. — Action sets are state dependent ω = (ωi)4i=1 ∈ {0, 1}4 with πi(ωi+1; ωi) = Pr(ωi+1|ωi). Allows us to enforce “trembles”. — Actions choices are ∗ If unshocked: ai(0) ∈ Ai(ωi = 0) = {R, S} ∗ If shocked: ai(1) ∈ {S} (early liquidation) • Shock processes: — Transitory: in each round of investment game, one agent receives shock, who receives it is iid across agents. — Permanent: in the Þrst round of investment game, one agent receives shock, who receives it is iid across agents.

• Payoffs. — Agent i’s payoffs from investment depend on one’s own actions and action choices of one’s neighbors, ui(ai, aj ), j ∈ N i(g) where ui(R, R) = a ui(R, S) = c ui(S, R) = b ui(S, S) = b ui(ai, ∅) = d if N i(gt) = ∅ • — In each round τ , receive average payoffs w.r.t. agents in your neighborhood: X

X

j∈N i ai∈Ai ,aj ∈Aj

µij ui(ai, aj ).

• Information. — Type (ωi) is private information. — Agents know the resulting network g, know those proposals made to them by j if they sent pij = 1, but not between other players. — Agents know the history of actions taken by their neighbors (aj , ∀j ∈ N i(g)).

Equilibrium i ∈ H i = Ω × P i × Γ × Ai × • ³ Agent i’s history h t t ´ j i ×j∈N i(gt)A × Ht−1 where Hti is the set of all possible histories for agent i.

• A behavior strategy of agent i is a history contingent plan of proposal and investment µ ¶ µactions de³

´

³

noted σi ∈ Σi = ×hi ∈H i ∆ P i × ×hi ∈H i ∆ Ai t t t t where ∆(X) is a prob. dist. over X. Let σ i(hit) = (ρi(hit), αi(hit)).

´¶

DeÞnition 1 A perfect Bayesian equilibrium is a strategy— ³ ´ i i −i i b b , β) such that (i), given β , σ b ∈ BR σ belief pair (σ , ∀i, t, hit and (ii), wherever possible, posteriors satisfy β i(hj ; hi) = prob(hj |hi).

b i is a best response in the • where, for instance, α investment stage if ∀αi

³ ´ ³ ´ i i j∈N (g ) i i i i j∈N (g ) i i t t i i b ,α v α ; ht, β ≥ v α , α ; ht, β

with expected payoffs for agent i given hit deÞned as: vi

µ

i αi, αj∈N (gt); hi , β i

X

X

t X



= j

j∈N i(gt) hj ∈H j {ait∈Ai (ωit),

+

t

X

t

j

j

β i(ht ; hit)µij (gt)ui(ait, at )αi

j

at ∈Aj (ωt )} µ ¶ µ i i (g j∈N (gt) i i i i i j∈N t q ht+1; ht, at, at v σ ,σ

i hit+1∈Ht+1 µ ¶ i (g ) j∈N t where q hit+1; hit, ait, at is a transition func-

tion for i giving the prob. next period’s history is hit+1 conditional on the current history and actions.

Predictions

• Two steps (work backwards). — For a given network structure, what do we predict about equilibrium investment play in a PBE? — Given the payoffs associated with step 1, what type of networks do we predict?

a+c > c and b ≥ d. Assumption 1 a > 2a+c > b > 3 2

• a > b > c is standard in coordination games • 2a+c 3 > b ensures that in UM, coordinated R play yields a higher payoff than S despite the fact that the shocked player is in one’s neighborhood. • b > a+c 2 ensures that if a shocked player is in one’s neighborhood in LI, R play is suboptimal. Necessary to get contagion started (related to allS play being risk dominant in a two player game).

• b ≥ d ensures participation is weakly optimal.

Proposition 1 If the shocks are transitory, network structure does not matter for investment actions.

• More formally, if the strategy in which each unshocked player chooses R in each period is a PBE for some network structure, then it is an equilibrium for any network structure.

• With i.i.d. shocks, there is nothing learned about the shock in one period that can be used to infer who will be playing S next period.

• In that case, the optimal ex-post investment strategy is the one that maximizes ex-ante payoffs.

• Since all realizations are equally likely, all unshocked agents receive 2a+c 3 in any network.

Predictions about Investment Play with “Permanent” Shocks

Lemma 2 In a UM network, there is an ex-ante payoff dominant, pure strategy PBE in which each unshocked agent plays R in every round. • Follows by 2a+c 3 > b.

Lemma 3 In an LI network, there is an ex-ante payoff dominant, pure strategy PBE in which all unshocked agents play R in the Þrst round, then all agents play S in subsequent rounds. • Follows by b > a+c 2 . • With round 1 all-R strategy, shocked player is discovered immediately.

• In round 2, agent diagonally across from the shocked agent anticipates S play by unshocked neighbors (and hence plays S).

• Very different if use naive best response. Implications for “speed” of contagion.

Lemma 4 In a M network, there is an ex-ante payoff dominant, pure strategy PBE in which all unshocked agents play R in the Þrst round, partners in the unshocked marriage play R in each subsequent round and partners in the other marriage play S in each subsequent round.

• R play in all periods by a pair of unshocked players yields the highest possible payoff a in each round.

• Since agents are ex-ante more likely to be in an unshocked marriage and strategy considered calls for S play to invoke an S response, in the Þrst round it is optimal to play R until one knows whether one’s partner is the shocked player, in which case it is optimal to play S since b > c.

Lemma 5 In an LI-UM network, there is an ex-ante payoff dominant, pure strategy PBE in which all unshocked agents play R in the Þrst round, all agents play S in subsequent rounds if a UM agent is shocked and all agents play R in subsequent rounds if an LI agent is shocked.

• Equilibrium has properties similar to that of either an LI or UM network depending on who gets the shock.

Lemma 6 In an LI-M network, there is an ex-ante payoff dominant, pure strategy PBE in which all unshocked agents play R in the Þrst round and then all agents play S in subsequent rounds.

• Equilibrium has properties similar to that of an LI network since the shocked player will be in some LI player’s 2 person neighborhood.

Predictions about Network Formation with Permanent Shocks Proposition 7 A UM network is not an equilibrium outcome.

• Suppose agent 1 deviates and chooses not to send a proposal to agent 2,while all other agents send three proposals. • Resulting LI-UM network means that agent 10s two neighbors (agents 3 and 4) ”provide insurance” to agent 1 (continue to play R) in the event that agent 2 gets the shock.

• In that event, agent 1 receives payoff a while in the UM network he would receive (2a + c)/3 < a.

• Agent 1 free rides.

Proposition 8 An LI network is an equilibrium outcome. • Suppose agent 1 deviates and chooses not to send a proposal to agent 4,while all other agents send two proposals • Resulting LI-M network is identical to equilibrium play in LI (M player, if he is unshocked, knows that one of the two LI players is linked to a shocked player after the Þrst round, thereby altering his beliefs and best responding with S play in the subsequent rounds as in lemma 3). • Since equilibrium play is the same, ex-ante payoffs are identical so that the deviation is not strictly proÞtable. • Not stable (in an informal evolutionary stability sense) to M proposal strategies.

Proposition 9 An M network is an equilibrium outcome.

• A unilateral deviation from sending a proposal to one’s partner results in autarky, where payoff dτ is strictly less under Assumption 1 than the exante payoff associated with M given by 2a+c 3 + 2a+b (τ − 1) . 3

Experiments on Contagion (Spread of S strategy) in Exogenous Networks • Without Shocks: — Timing: 10 rounds of investment actions in P1 with a+c 2 > b (so all-R is payoff and “risk” dominant),10 rounds in P2 with a+c 2 < b (where all-R is payoff dominant and all-S is “risk” dominant): P1 (i, j) R S R 60 20 S 35 35

P2 (i, j) R S R 60 0 S 35 35

• With Shocks: — Timing: Shock, 5 rounds of invesment in P1, Shock, 5 rounds in P1 (to get experience with shocks), Shock, 5 rounds in P2, Shock, 5 rounds in P2.

Hypotheses in Exogenous Networks (from Lemmas)

• Without shocks: if coordinate on payoff dominant all-R equilibrium in P1, then remain in it in all networks in P2.

• With shocks: if coordinate on payoff dominant all-R equilibrium in P1, remain in all-R in UM and unshocked M, not in LI.

Experimental Results

• Without shocks: UM (3 yes), LI (1 yes, 1 no), M (4 yes).

• With shocks: UM (2 yes, 1 no), LI (2 yes), M (6 yes).

Experiments with Endogenous Network Formation

• All payoffs as in Assn 1 (P2) • Timing: — 1x Impose network (UM, LI, M to help with proposal coordination) followed by shock and then 5 rounds of investment actions. — 4xNetwork proposals followed by shock and then 5 rounds of investment actions.

Hypotheses in Endogenous Networks (from Props)

• M is stable, LI is weakly stable, UM is not stable Experimental Results

• M (3 yes), LI (3 no), UM (3 no).

Conclusions

• Need more experiments to conduct a serious data analysis (quantile response)

• Evidence for contagion in incomplete (LI) networks.

• Evidence for “small is beautiful” (M) network choice on the basis of ex-ante payoff dominance (i.e. M>UM>LI, note nonlinearity as well), but interesting ex-post issues (UM>M|shocked).

• Free Rider problem in UM network=⇒Government Intervention in Banking Networks?

Extensions

• Different shock process: no shock with probability q and permanent shock with prob. 1 − q. Can we get risk dominant, contagious behavior to kick in more frequently?

• Renegotiate proposals? Ex-post an unshocked M player would like to match get into LI with the other two unshocked partners so possibility of side payments.

Experiments with "etwork Economies

Each agent sends link proposals to all other agents p i. D (p i j. ) 4. jD1 e н0,18. 4 . ( Mutually agreed upon links are formed (i.e. link ij occurs iff p i j p j i. D 1). ( A network is 4ust the set of all agreed upon links (g D нij : Vi, j e m8 e Γ, where Γ is the set of all possible networks). ( For simplified version, network formation is.

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