Cycles of Aggregate Behavior in Theory and Experiment

Games and Economic Behavior 36, 105–137 (2001) doi:10.1006/game.2000.0821, available online at http://www.idealibrary.com on

Cycles of Aggregate Behavior in Theory and Experiment1 Antoni Bosch-Dom`enech Universitat Pompeu Fabra, 08005 Barcelona, Spain

and Mar´ıa S´aez-Mart´ı Stockholm School of Economics, S-113 83 Stockholm, Sweden Received December 30, 1998; published online April 26, 2001

We test in the laboratory the potential of evolutionary dynamics as predictor of actual behavior. To this end, we propose an asymmetric game (which we interpret as a borrower–lender relation), we study its evolutionary dynamics in a random matching setup, and we test its predictions. The theoretical model provides conditions for changes in qualitative aggregate behavior in response to variations in structural parameters. While it turns out that Nash equilibrium is not a reliable predictor of average aggregate behavior, the experiment seems to confirm the qualitative predictions of the evolutionary model under structural changes. Journal of Economic Literature Classification Numbers: C7, C9, E3. © 2001 Academic Press Key Words: cycles; evolutionary dynamics; games; experiments.

1. INTRODUCTION Evolutionary game theory has been a hot topic of research in recent years and an important body of results is now available (see Weibull, 1995). Yet, 1

We thank Albert Satorra, Fabrizio Zilibotti, the Associate Editor, and two referees for their very helpful comments, and Xavier Herrero for his assistance in programming the experiments. Financial support from the Spanish Ministry of Education and Culture under Contracts SEC98-1853-CE and PB98-1076 and from EMU-TMR Research Network ENDEAR (FMRX-CT98-0238) is gratefully acknowledged. Mar´ıa S´aez-Mart´ı thanks the Jan Wallander and Tom Hedelius’ Foundation for financial support. 105

0899-8256/01 $35.00

Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.

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experimental evidence about the relevance of the use of evolutionary dynamics in real world economic situations is still limited (see, e.g., Friedman, 1996). This paper studies theoretically, and tests experimentally, the evolutionary dynamics of a two-population asymmetric game in which one of the populations has an outside option. In our model, players from two different populations, which we shall call lenders and borrowers—who know nothing about each other’s individual history—are randomly matched. Players in the first population (lenders) choose to play the outside option (invest in a safe asset) or to continue the game (grant a loan). In the latter case lenders and borrowers play a 2 × 2 simultaneous move game with a unique mixed equilibrium. In this subgame, borrowers decide whether to exert effort or not. Effort, which is unobservable, positively affects the probability of success and repayment. Lenders can detect shirking and enforce a punishment, but only if they engage in a costly (interim monitoring) process. If lenders decided to monitor all projects, the borrowers’ payoff would be maximized by exerting effort, since shirking would certainly be detected and penalized. However, if no borrower shirked, lenders would maximize profits by not monitoring any loan. The model has two sets of Nash equilibria. One equilibrium characterized by a mixture of good and bad behavior on the borrowers’ side, and with lenders randomly monitoring a positive proportion of the loans granted. We interpret this equilibrium as a fully developed credit market since, in this case, all matches between borrowers and lenders result in a credit relation. The other equilibrium (or, more exactly, set of equilibria) is characterized by complete “financial collapse.” When lenders expect to find many bad borrowers they prefer to invest in the safe asset (play the outside option) and do not grant any loans. If the proportion of borrowers who would cheat—should they receive a loan—is large enough, the lenders behave optimally by not lending. The main focus of the theory is the characterization of the dynamics under the assumption that agents follow an adaptive behavior. We assume that individuals from large populations (borrowers and lenders) are randomly matched to play repeatedly a one-shot game representing the credit relation just described. In this framework, Nash equilibria are viewed as stationary points of dynamic processes representing some kind of evolutionary adaptation. We shall assume a rather general class of dynamics whose only requirement is that strategies with higher payoffs grow relative to those with lower payoffs. These “payoff monotonic dynamics” can be obtained from models of imitation and learning, with individuals revising their strategies in the light of the different strategies’ relative payoffs. In the presence of this adaptive behavioral rule, cyclical patterns of credit and effort are an intrinsic feature of the impulse–response function of an

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economy subjected to shocks. In particular, when a shock hits the economy, reducing the probability of successful investments, potential lenders become increasingly “scared” at the growing number of defaults which they observe and start switching out of loans into the outside option. As a result, the economy is temporarily driven away from the good and toward the bad equilibrium. However, before the complete financial collapse is reached, granting credit to borrowers may turn profitable again and the economy starts reverting toward the good equilibrium in which all applications for a loan are satisfied. The emergence of cycles relies on the assumption that agents are not fully rational. If agents played Nash equilibrium strategies at any moment in time, we could only observe either a fully active or a missing credit market. But as out-of-equilibrium behavior is part of the actual play, the cycles describe the dynamic behavior of populations who learn their way to equilibrium through iterative play. The dynamics postulated in the paper predict the existence of two sets to which the economy can converge in the long run: (a) the set of Nash equilibria described as credit collapse and (b) the set of states described as a fully developed credit market which contains the (mixed) Nash equilibrium characterized above. The model also gives exact predictions of when changes in aggregate behavior, which we describe as credit cycles, will take place. We run the experiment (i) to determine whether there is credit activity in the long run, (ii) to test whether Nash equilibrium is a good predictor of behavior, and (iii) to test whether credit contractions occur as predicted by the model. Concerning (i), the experiment clearly rules out the credit collapse set of equilibria and, in most cases, shows convergence to the set with full credit availability. Concerning (ii) the experiment shows that Nash equilibrium is not a reliable predictor of average behavior. This last result contradicts neither our theoretical model, since convergence to the mixed Nash equilibrium is not guaranteed under general monotonic dynamics, nor other experimental results which show the difficulty in obtaining convergence to mixed equilibria in two populations games (see Friedman, 1996). Finally, concerning (iii) the experiment shows that cycles tend to occur as predicted by the model. Our analysis relates to various streams of literature. Although our interpretation of the game as a borrower–lender relation is more pedagogical than substantive, the emergence of periodical episodes of credit rationing and their effects on the aggregate economic activity have been widely discussed in the macroeconomic literature. In this literature, the emergence of credit cycles is typically due to changes in the value of net worth and collateral in the hand of the borrowers (see Bernanke and Gertler, 1990, and

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Kiyotaki and Moore, 1997). Instead, in our game-theoretical model, cycles (of credit) emerge from simple evolutionary dynamics.2 The paper is related to the vast recent literature on learning and the use of experimental data to fit parametric learning models. Erev and Roth (1998) argue that their reinforcement models fit the data better than fictitious play type models or best-response models of learning. Van Huyck et al. (1997) use a model of logistic fictitious play as a learning model and show that it fits their data quite well, while reinforcement learning predictions are not observed. Cheung and Friedman (1997) report that weighted fictitious play models do better at fitting the data than reinforcement learning type models. Camerer and Ho (1999) build and estimate a more general learning model that contains elements of both reinforcement learning and belief learning, which ensues in better fits than either reinforcement learning models or belief learning models. Empirical tests of replicator dynamics (Cheung and Friedman, 1998) result in worse fits than models of belief learning. Given the difficulty in distinguishing different learning models from experimental data (see also Tang, 1996), and the somewhat disappointing tests of replicator dynamics, we do not intend to use experimental data to fit a specific evolutionary model. Rather, we want to test how well payoff monotone dynamics accounts for empirical data under changes in structural parameters. Using the relative analytical simplicity of evolutionary models to make precise aggregate qualitative predictions, we check the fit to experimental data of these predictions, when some basic parameters change. In other words, we are concerned by how qualitative behavior depends on structural parameters (payoffs). From this perspective, the paper shares a common purpose with the analysis of bifurcations of equilibria in continuous-time dynamical systems (see, e.g., Wiggins, 1990, or Kuznetsov, 1995). The paper is organized as follows: In Section 2 we describe the model, characterize the equilibria, and study the dynamics of the economy. In Section 3 we describe the experimental setup and the different treatments of the experiment. In Section 4 we present the results of the experiment and test the predictions of the model. Section 5 concludes. 2 The major limitations of our model, as a description of credit cycles, lie in its partial equilibrium nature (interest rates are exogenous) and the lack of long term relations between borrowers and lenders. Nevertheless, the model could be regarded as a stylized description of a market for loans to new investors and small firms, whose access to the credit market is sporadic and results in unreliable information about their past behavior. Bernanke (1983) observes that this segment of the market was significant and important during the Great Depression, a major episode of prolonged credit crunch, when customer relations were weakened by the fact that many borrowers were separated from their banks—when these had to close—and forced to ask for credit in banks that were new to them.

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2. MODEL We consider a stylized economy in which all agents are risk-neutral. There exists a safe asset in the economy that earns an exogenously given interest rate. All potential investors need to borrow a fixed amount of money W to finance a project and have no collateral. Borrowers may either exert effort or shirk. Each project can have two outcomes (states), good or bad, and outcomes are publicly observable at the end of each period. Effort increases the probability that an investment is successful and has a good outcome. In the good state the investment is successful and the borrower gets revenue H > W , cancels the debt with the agreed payment R, and earns a net profit, whereas in the bad state the borrower gets nothing and cannot repay the debt.3 Let π and π + α be the probabilities of the good state without and with effort, respectively and let e be the disutility of the effort. Lenders can decide to grant a loan of size W or to invest the same amount of money in a safe asset which yields a gross return r. Moreover, lenders are entitled to interim monitoring of borrowers’ activity. By monitoring, the lender will be able to observe the effort exerted by the investor. If cheating is detected, the lender asks for his or her money back (without earning interest) and the borrower is liable to legal prosecution, with a nonpecuniary utility loss f . If no cheating is detected the project can continue. However, monitoring entails a cost c (as if the lender had to pay some specialized institution for this purpose). Figure 1 is the extensive form representation of the situation described above. The lender decides whether to invest in a safe asset (SA) or to grant a loan (L). If the loan is granted, the borrower decides either to exert effort (E) or to shirk (S) and the lender chooses between no monitoring (NM) and monitoring (M). Observe that the lender has four pure strategies: SA NM, SA M, L NM, and L M.4 Since the strategies SA NM and SA M are behaviorally indistinguishable, we will refer to both of them with the same label SA. For notational simplicity we will relabel L NM and L M as NM and M, respectively. Figure 2 is the normal form of the borrower–lender game.

3

We will treat R as exogenous, although this can be regarded as the outcome of a more complicated contractual relation. In particular, we implicitly assume that there is no equilibrium value of R that makes the borrower’s choice incentive compatible, i.e., that induces the borrower to exert effort, while giving the lender a higher expected payoff than that warranted by investing in the safe asset. 4 Alternatively we could write an extensive form with the lender having a single decision node with three actions, safe asset, monitoring, and no monitoring. Both extensive forms will lead to the same “reduced” normal form game.

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110

FIG. 1. Borrower–Lender game.

2.1. Equilibria Let us assume that players can randomize over pure strategies. In the borrower–lender game the strategy spaces are S1 = SA NM M and S2 = E S. Let player i’s mixed strategy space be denoted by Si  and let xih be the probability assigned by the mixed strategy xi ∈ Si  to player i’s hth strategy. Let x11 , x12 , and 1 − x11 − x12 be the probabilities assigned by a lender to strategies SA, NM, and M, respectively and let x21 and 1 − x21 be the probabilities assigned to the strategies E and S by a borrower. We shall describe a mixed strategy profile by the vector x = x11  x12  x21 . We assume that payoffs satisfy the following conditions: (c.1)

π + αR − c > r > W − c > πR.

(c.2) W − c − πRπ + αR − W  > r − W − cW − πR. (c.3)

π + αH − R − e > f .

(c.4) αH − R − e < 0.

FIG. 2. Normal form game.

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Condition (c.1) guarantees that when the borrower exerts effort, the lender’s most profitable strategy is NM followed by M and SA, whereas when the borrower shirks the ordering is strictly reversed. Conditions (c.1) and (c.2) together guarantee that every strategy in S1 is a strict best-reply for some values of x21 . Condition (c.3) guarantees that, when the lender monitors, exerting effort is the best-reply for the borrower. Condition (c.4) guarantees that, when the lender does not monitor, to shirk is the bestreply for the borrower. Notice that the optimal strategy of the borrowers depends only on the ratio of nonmonitored loans over total loans, x12 /1 − x11  ≡ xˆ 12 , since the payoff to E and S is the same when the lenders invest in the safe asset. If payoffs satisfy conditions (c.1)–(c.4), the Borrower–Lender Game has a set of Nash equilibria with two components: (i) x∗12 = (ii) where

a mixed equilibrium x∗ = 0 x∗12  x∗21 , where π + αH − R − e − f πH − R − f

and

x∗21 =

W − c − πR

W − πR

(1)

a set C = x1  x2  ∈ S1  × S2  x11 = 1 and x21 ≤ x¯ 21 , x¯ 21 =

r − W − c  π + αR − W

(2)

Observe that (c.2) guarantees that x¯ 21 < x∗21 . It is important to notice that SA is the lenders’ best strategy when the proportion of borrowers who exert effort is below x¯ 21 . This critical threshold will play a crucial role in our characterization of the dynamics, since it will allow us to determine the conditions under which cycles will emerge. The mixed equilibrium, which is subgame perfect, corresponds to the existence of credit. All equilibria in C imply no credit, namely all rowplayers choose the outside option. Under our restrictions on the payoffs, any equilibrium belonging to C, which corresponds to the absence of credit, is Pareto dominated by the singleton equilibrium. The problem of the existence of credit therefore reduces to studying an equilibrium selection problem in game theory. In the following section we follow an evolutionary approach. 2.2. Dynamics Rather than assuming that we have two players randomizing over pure strategies, we assume that there are two large populations of boundedly rational players, playing pure strategies, which are randomly matched. In this view a mixed strategy in population i is a population profile xi ∈ Si 

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with xih ≥ 0 denoting the relative frequency of the hth pure strategy in population i. We assume a very general type of continuous dynamics, payoff monotonic, which ensures that more profitable strategies increase relative to less profitable strategies. This type of dynamics can be obtained from models of imitation and learning (Friedman, 1992; Weibull, 1995).We could assume, for instance, that players can observe a sample of contemporaneous interactions and imitate more profitable strategies. Lenders could meet and talk about businesses, and borrowers tell each other their credit experiences. We describe evolution of the state of the economy, x, by a system of differential equations (time indices suppressed) x˙ ih = xih gih x

∀ i ∈ 1 2 h ∈ Si  x ∈ ×Si  ≡ 

(3)

where gih is the growth rate of pure strategy h in population i.5 Under these dynamics, both  and its interior int() are invariant, and extinct strategies stay extinct forever. Under payoff monotonic dynamics, πih x > πik x

⇐⇒

gih x > gik x

(4)

for all h k ∈ suppxi , where πih x is the expected payoff to a player from population i who employs strategy h in state x. Notice that the ordinal relationship applies only to nonextinct strategies; for extinct strategies gih x = 0. Since several strategies will coexist at any time, including strategies which are not current best-replies and agents are randomly matched, the expected payoff to each strategy will depend on the probability of matching with each of the strategies played by the opponent population. In a world with many honest borrowers, to lend without monitoring is likely to be a successful strategy. In a world with almost all dishonest borrowers the best one can do is to invest in the safe asset. The state space, with the Nash equilibrium components, is represented in Fig. 3. The states where the credit market is fully developed (x11 = 0) correspond to the floor of the polyhedron, which contains the mixed equilibrium. The set of Nash equilibria with credit collapse is the thick segment, C, on the edge where x11 = 1. The arrows along the edges of the polyhedron show the associated directions of the vector field for any monotonic dynamics. We have also drawn the directions of the vector field on the face where x11 = 0. Under payoff monotonic dynamics, the behavior in the face F = x ∈  x11 = 0 (the floor in Fig. 3) can be of three types: (1) convergent
We assume, as is standard, that (i) g is Lipschitz continuous on  and (ii) h∈Si x˙ ih = h∈Si xih gih x = 0 ∀ i ∈ 1 2 x ∈ ×Si . Existence and uniqueness of a solution are guaranteed by the Picard–Lindel¨ of theorem.


5

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FIG. 3. State space and equilibria.

to the mixed equilibrium, (2) closed orbits around the equilibrium point, and (3) convergent to the boundary of F. Figure 4 represents the different dynamics on the face F (i.e., with x11 = 0), the mixed equilibrium x∗ , and the value x¯ 21 . On the horizontal dimension we plot the proportion of nonmonitored loans and on the vertical the proportion of borrowers who exert effort. The dashed line is the critical value x¯ 21 . In all three cases the dynamics are clockwise. To fully characterize the dynamics we need to study the stability properties of the sets C and F. We assume that some players tremble, mutate, or experiment. Alternatively we could assume that new players, who know nothing about the economy, come along and play arbitrary strategies. The trembles’ role is to resurrect extinct strategies and to perturb the rest points of the dynamics. The absorbing set C is not a Lyapunov stable set. To prove this result we use the fact that no Nash equilibrium that belongs to a nonsingleton component of Nash equilibria is asymptotically stable in any payoff-monotonic dynamics (see Weibull, 1995) and show that one element of C is not Lyapunov stable; hence C as a whole is not Lyapunov stable. Note, however, that if the system is perturbed starting in C  = x1  x2  ∈ S1  × S2  x11 = 1 and x21 < x¯ 21  ⊂ C, the dynamics converge back to C since x˙ 11 > 0 whenever x21 < x¯ 21 and x11 ∈ 0 1.

FIG. 4. Dynamics on face F.

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114 Proposition 1.

The set C is not Lyapunov stable.

Proposition 2. For any payoff monotonic dynamics, the face F is reflecting for x21 ≤ x¯ 21 and absorbent for all x21 ≥ x21 , with x¯ 21 = Proof.

r − πR r − W − c < = x21 < x∗21  π + αR − W αR

(5)

See the Appendix for a proof.

Conditions (c.1) and (c.2) guarantee that investing in the safe asset is the lenders’ best strategy for efforts below x¯ 21  and it is the worst strategy for efforts above x21 = r − πR/αR with x¯ 21 < x21 < x∗21 . Under payoff monotonicity, “mutations” involving SA will spread if they occur when x21 ≤ x¯ 21 and will be killed off when x21 ≥ x21 . The behavior for intermediate values of x21 depends on the particular specification of the dynamics. Observe that as the proportion of SA increases (credit contraction), so does the proportion of monitored loans over total loans, since, due to the payoff monotonicity and assumptions (c.1) and (c.2), M grows relative to NM when x21 ≤ x∗21 and the advantage of shirking over exerting effort gradually disappears. The process of credit contraction will be reverted if a state with x21 = x¯ 21 is reached. The following proposition establishes that there are interior orbits that exhibit credit cycles, namely periods of credit contraction (increase of SA) followed by periods of credit expansion (decrease of SA). 1 = x ∈ int x21 = x¯ 21 and x12 < Proposition 3. For all x ∈ H 1 − x11 x∗12  the orbit through x has at least one credit cycle. Proof.

See the Appendix for a proof.

Figure 5 shows a simulation with replicator dynamics6 which illustrates the stability properties of the set C and the face F, as well as the existence of credit cycles. Under replicator dynamics all strategies which get a higher than average payoff have positive rates of growth and those with higher payoff grow faster. The mixed equilibrium is a center point which is neutral, neither asymptotically stable nor unstable (Case 2 in Fig. 4) (see Hofbauer and Sigmund, 1988). 6

Under replicator dynamics the rate of growth of population share xih is given by gih x = πih x − π¯ i x

∀ i ∈ 1 2 h ∈ Si x ∈ ×Si 

where π¯ i x is the average payoff in population i. Justifications of the use of the replicator dynamics in the modelling of learning are found in Cabrales (2000), B¨ orgers and Sarin (1997), and Schlag (1998). See Samuelson and Zhang (1992) for the properties of replicator dynamics which are shared by monotonic dynamics.

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FIG. 5. Stability properties of C and F.

The system is initially in state a in C. A mutation (with high enough proportion of monitored loans) drives the state out of the set of Nash equilibria, since effort starts growing and so does credit availability. Observe that the orbit reaches states in F (such as b in Fig. 5) with a high proportion of good borrowers and nonmonitored loans. Under any monotonous dynamics, both S and NM grow: in a population with many good borrowers from the lenders’ point of view, it is better to grant loans without paying the cost of monitoring, while from the borrowers’ perspective, cheating is better since the risk of being caught is small. The system will move East and reach states characterized by a high rate of cheating (grey area). In those states the outside option turns out to be relatively profitable (state c in the figure) and the system falls into a progressive credit reduction. Notice, however, that this process does not lead to the complete disappearance of credit activity and the economy reverts to a new stage of credit expansion. An implication of the previous result is that shocks can generate cycles. For instance, a shock that reduces the probability of success tends to generate cycles since its effect is to increase x¯ 12 and to make the economy more vulnerable to the invasion of SA. Assume, for the sake of simplicity, that the economy is initially in the mixed equilibrium and that a shock which changes the payoff matrix and the equilibria, although not their topological properties, hits the economy. The initial equilibrium is no longer stable and we can observe cycles as illustrated in Fig. 6, which shows a simulation in which the initial equilibrium a becomes unstable after the shock. Credit cycles occur before the economy reaches a new long run equilibrium with higher proportion of monitored loans and of ‘good’ borrowers. Observe that the new mixed equilibrium is asymptotically stable in the face F. Notice that we have generated cycles with dynamics which

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FIG. 6. Postshock behavior.

are “convergent” in the face F. These dynamics are the “least favorable” for generating cycles, since we cannot guarantee that values below x¯ 21 are reached (see Case 1 in Fig. 4). 2.3. Summary of the Theoretical Results In the previous subsection we have studied the behavior of the economy under very general evolutionary dynamics. We have established the existence of two absorbing sets: (i) a set of Nash equilibria that we interpret as credit collapse (set C) and (ii) the set of states that we interpret as a fully developed credit market (set F) which contains a unique (mixed) Nash equilibrium. In this equilibrium there is a mixture of good and bad borrowers, and the lenders, who grant all possible loans, monitor a positive proportion of them. We have also shown that there exists a critical value, a proportion of borrowers exerting effort x¯ 21 below which lenders prefer to invest in the safe asset. If such or a lower proportion of good borrowers is reached, a drainage of funds from loans into safer investments will be observed. An implication of this result is that shocks that occur when the economy is in the absorbing set that we interpret as a fully developed credit market can generate credit cycles, initiated by a credit contraction followed by a credit expansion. In what follows we present the results of an experiment which reproduces in the laboratory the situation described in the model. The experiment will allow us to select one of the two absorbing sets and to test empirically the predictions of our theoretical model about cycles.

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3. EXPERIMENT 3.1. Experimental Setup 3.1.1. Generalities As in the theoretical model, there exist two sets of experimental subjects7 , A and B. Each period, a random pairwise procedure matches individuals (whose identities are kept confidential) in both sets, approximating a series of two-player nonrepeated games. An individual in A, a lender in our interpretation, can each period either invest in a low return r, safe asset (SA) or lend, at a fixed return R, to the individual in B, a borrower, with whom he or she has been matched. Since the identity of the two matched individuals is unknown to them, the interest rate charged for the loan cannot be made dependent on the borrower’s history, and it is assumed to be always the same. Now, lenders can choose to monitor (M) the use of the credit to detect possible fraudulent behavior, but this monitoring entails a cost c. Borrowers, on the other hand, can either “shirk” (S) or exert “effort” (E). The experiment consist of a sequence of 13 laboratory sessions (without preliminary pilot sessions), lasting from 60 to 90 minutes each, using profit motivated subjects. Payoffs are calibrated to result in average earnings of about Peseta 2,000 (about $14) per hour per subject. Realized earnings depend on chosen actions and typically vary from Peseta 500 (US $4), to Peseta 3,500 ($27) per session. All subjects are given written instructions (available from the authors on request) in a neutral language that does not mention the interpretation of their roles as borrowers or lenders, nor does it include any reference to lending, consuming, or investing. The instructions contain information about the mechanics of the experiment and about the subject’s payoffs, which depend, as is made explicit, on her decisions and the decisions of the matched pair. In the instructions, subjects are not informed about payoffs of any other subjects. Previous to the beginning of the sessions, subjects receive about 15 minutes training on the computer. In each session, 12 or 14 undergraduate subjects interact for an undisclosed number of periods.8 All sessions have in common that, in each period, the subjects, seated at visually isolated terminals, review historical data and take actions from a 7

The description follows the usual protocol, as in Friedman (1996, p. 10). We intended to use 14 subjects in all sessions. Failure of a computer in one session and failure to attract the required 14 subjects in another resulted in 2 sessions (10 and 11) with only 12 subjects. Additionally, to see whether our averages were affected by the small number of observations, in Sessions 3, 9, 10, and 11, we forced subjects to make 3 choices instead of 1 per period, as if we had 42 subjects participating. We fail to see any effects of this procedure. We kept sessions to a minimum of 32/34 periods, and some sessions went up to 62 periods. 8

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menu of two possible actions for borrowers (which we interpret as effort (E) and no effort (S)) and three possible actions for lenders (which we interpret as invest in the safe asset (SA), lend without monitoring (NM), and lend with monitoring (M)). The choices of all subjects are sent to a central processor that computes the outcomes. The historical data that appear on their screens, after each period, are their previous decision and payoff and the previous decision and payoff of another subject, chosen randomly, in their group (either A or B).9 In all sessions, the payoff function is defined for a set of parameters that satisfy the conditions (c.1)–(c.4) of the theoretical model. The abstract payoff matrix used is the same as in the normal form game showed in Fig. 2, but subjects only see as the elements of the matrix the expected payoffs that result from the operations stated there. These payoffs are the same for each group of subjects. In most sessions, a shock occurs in midsession resulting in a change in the payoff matrix.10 The remaining sessions (1, 5 and 9) do not include a shock and, consequently, the payoff matrix stays the same during the whole session. When a shock exists, the payoffs are chosen to separate as clearly as possible the preshcok and postshock mixed Nash equilibria predicted by the model.11 3.1.2. Treatment The treatment is the payoff: We want to observe the effects of changes in payoffs. We organize the experiment in three groups of four sessions, each with a different starting payoff matrix. Among the four, one session, serving as a baseline, does not include a shock, or payoff change, in midsession. These shocks originate in a modification of one or several of the parameters 9

We wanted to smooth out the possible impact of extreme information treatments (no information or full information). For this reason, and also impelled by the interpretation of the game as a lender–borrower game, we settled for an intermediate information treatment: Participants are partially informed of what others do. In addition, by this procedure, the probability of sampling a strategy equals its current share in the population. This proportional sampling is quite common in justifying, in terms of social learning, the use of payoff monotone dynamics in economics. 10 In Sessions 2, 6, 7, and 8, the shock was announced and the new payoff matrix was made public. In the remaining sessions with shocks, the shock was not announced and the new payoff matrix was never made public, but subjects were told that the payoff matrix would change during the session without warning. We have failed to see any systematic difference between the two treatments. In particular we fail to see, as one would have expected, more experimenting on the subjects’ part when the new payoff matrix was not explicitly given. These changes, plus the ones reported in footnote 6, were introduced to check for robustness, as we replicated the sessions with these variations. 11 In Sessions 1, 2, 3, and 4, the equilibrium values before and after shock for NM were considered to be too close, and the payoffs were subsequently changed so that the distance between equilibria increased.

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TABLE I Summary of the Sessions’ Basic Features Group 1 1 1 1 2 2 2 2 3 3 3 3 —

Session

Payoffsa

Periodsb

Number of Subjects

1 2 3 4 5 6 7c 8 9 10 11 12 13

P1 P1/P2 P1/P2 P1/P2 P3 P3/P4 P3/P4 P3/P4 P5 P5/P6 P5/P6 P5/P6 P6/P5

32 17/32 17/32 26/62 50 20/50 21/50 23/50 30 17/34 17/34 16/62 17/34

14 14 14 14 14 14 14 14 14 12 12 14 14

a The payoffs column indicates the different payoffs used in the experiment. A slash means that a shock occurred in the session, resulting in different initial and final payoffs. Sessions 1, 5, and 9, acting as baselines, do not have any shock. Note that, in Session 13, the order of payoffs was reversed with respect to Group 3 sessions. b When the number of periods is separated by a slash, the first figure indicates the period when the shock occurred, and the second the total number of periods. c Session 7 includes a cost of changing strategies with the purpose of reducing the volatility of the market.

that affect the expected payoff—such as the probability of success with or without effort—in the normal form game matrix. In addition, a Session 13 was organized to observe the effect, if any, of reversing the order of the payoff matrices with respect to Sessions 10, 11, and 12. Table I summarizes the experimental setup. Our main goal is to test whether, as predicted by the payoff monotonous dynamics of the model, credit contractions occur at the critical threshold. We also want to check whether Nash is a good predictor of average behavior and whether the observed dynamics show any degree of convergence toward Nash. Finally we use the experiment to discriminate between the two sets of equilibria of the game.

4. EXPERIMENTAL RESULTS 4.1. Payoff Salience Since the experiment involves several sessions, we want to check that they take place in a stable experimental setting. To this end, we compare sessions with the same initial payoffs, looking at the average frequencies

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TABLE II Table of Frequenciesa Payoff P1

Payoff P2

Payoff P3

S

x11

xˆ 12

x21

S

x11

xˆ 12

x21

S

x11

xˆ 12

x21

1 2 3 4

0.12 0.12 0.14 0.10

0.58 0.68 0.64 0.65

0.69 0.74 0.73 0.63

0.55 0.49 0.44 0.26

0.70 0.05

0.25 0.49 0.60 0.49 0.51 0.47 0.13

0.35 0.29 0.54 0.48

0.62 0.07

0.56 0.80 0.78 0.78 0.74 0.73 0.10

0.29 0.27 0.28 0.26

0.12 0.02

0.21 0.09 0.09 0.09 0.07 0.11 0.06

9 10 11 12

Avg SDV

5b 5b 6 7 8 Avg SDV

Avg SDV

0.28 0.01

0.48 0.12

0.26 0.13

a Sessions (S) grouped according to payoffs. First part of sessions: x11 (safe asset); xˆ 12 (nonmonitored loans over total of loans); x21 (effort). The equilibrium values of the singleton equilibrium, x∗11  x∗12  x∗21 , for the payoffs matrices P1, P3, and P5 are (0, 0.93, 0.54), (0, 0.79, 0.22), and (0, 0.4, 0.71), respectively. b The original Session 5 was interrupted short by a computer failure. So we decided to run it again. We call 5 the new session and we label the old session as 5b. The peculiar results in Session 5 may be due to these disturbances.

of the different strategies before any shock occurs. As for those sessions without shock, we look at the average frequencies for the same number of periods as in the other sessions. In broad terms—see Table II—average frequencies of SA (x11 ) and nonmonitored loans over total loans (xˆ 12 ) remain more similar within groups than between groups, confirming some basic stability of the experimental setup. They also indicate that payoffs have a salient effect on outcomes. Specifically, if we leave aside Session 5, which does not fit squarely in any of the three groups, SA has frequencies in the range 10–14% for P1, in the range 7–9% for P3, and in the range 26–29% in P5. Similarly we observe that xˆ 12 frequencies are in the range 52–65% for P1, in the range 74–80% for P3, and in the range 29–48% for P5. Turning to borrowers, frequency differences among groups for the effort strategies are less marked, but still clearly discernible. P1 has frequencies in the range 63–74%, P3 in the range 49–60%, and P5 in the range 26–55%. The conclusion is that stability of the experimental environment and payoff salience seem to be verified. 4.2. Which Absorbing Set? The game presented in the theory part of the paper has two differentiated sets of Nash equilibria. One is a set of equilibria with credit collapse; the second is a full credit situation with some loan monitoring and effort. In the experiment the situation of credit collapse was never reached. The worst case represents a credit reduction of about 75% of the credit potential, which

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121

occurs in one period of Session 4 and in one period of Session 12. In the experiment, a “serious” credit contraction meant usually a credit reduction of 50–60%, a long way from the credit collapse. Conversely, situations with full credit availability abound and, as we will see below, convergence to full credit availability is not uncommon. The first conclusion of the experiment is, therefore, to question the likelihood of the equilibria that we describe as credit collapse. This set of equilibria, C, does not seem to be a good asymptotic description of the long run behavior of the system, in contrast to what was suggested, for a similar unstable set of equilibria, by Binmore and Samuelson (1999).12 4.3. Convergence to Full Credit Availability Our general model claims attractor capability for the full credit strategy. For the claim to be substantiated by the experiment we should observe a tendency to go to the floor of Fig. 3, since this floor represents no investment in the safe asset. We now evaluate whether observed behavior converges to the fully developed credit market. We say that state x converges to x* at time t   with a preselected tolerance bound b if T 1  xt − x∗  ≤ b T −t t

∀ t ≥ t

(6)

where T is the last period. When sessions have shocks, we run the test for the two subsessions. Following Friedman (1996), we choose b = 1/n (tight convergence) and b = 2/n (loose convergence), where n is the number of subjects in each set A and B.13 Observe that the tight criterion will yield convergence from period t onward as long as no more than one subject deviates, on average, from the equilibrium strategy, while the loose criterion allows for, at most, two deviating subjects on average. Table III reports the convergence results toward full credit availability x11 = 0. In the preshock part, out of 13 sessions, 5 show strong convergence to full credit availability from the very beginning; a total of 8 show strong convergence from some period onward; a total of 9 show some sort of convergence; and only 4 show no convergence. In the postshock part, out of 10 12 Binmore and Samuelson (1999) suggested that, near a set of unstable equilibria, deterministic drift could be large enough to force the system to converge to these equilibria. It can be argued that, in our experiment, no session was long enough to justify talking about the long run, nor was the system ever close to the unstable equilibria. 13 Our criterion for convergence differs from Friedman’s in that we check for convergence from every period onward, while he does it for the whole session (or half session). As explained in footnote 8, n = 7 except in Sessions 3, 9, 10, and 11, when n = 21.

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122

TABLE III Convergence toward Full Credit Availability x11 = 0a Session 1 2 3 4 5 6 7 8 9 10 11 12 13

Periods

Preshock

Periods

1–32 1–16 1–16 1–25 1–52 1–19 1–20 1–22 1–30 1–16 1–16 1–25 1–16

T(1), L(1) T(1), L(1) T(16), L(15) T(1), L(1) No convergence T(19), L(1) T(1), L(1) T(1), L(1) L(30) No convergence No convergence No convergence T(16), L(1)

— 17–32 17–32 26–62 — 20–50 21–50 23–50 — 17–34 17–34 26–62 17–34

Postshock — T(22), L(17) No convergence T(26), L(26) — T(50), L(20) T(21), L(21) T(46), L(23) — No convergence T(31), L(25) T(34), L(26) L(23)

a T(t) and L(t) stand respectively for tight and loose convergence from period t onward. Of course, T(t) implies L(t).

sessions, 7 show strong convergence; another 1 shows loose convergence; and only 2 sessions show no convergence. We interpret this result as implying behavioral convergence toward the full credit equilibrium, as predicted by the payoff monotonic dynamics of our model. 4.4. Nash as an Approximation As we saw in Section 2.2, payoff monotonic dynamics do not guarantee convergence to the mixed equilibrium. But is the Nash equilibrium (with full credit availability) a good approximation of the average frequency of strategies? Table IV compares the theoretical frequencies of the strategies at this Nash equilibrium with the average frequencies observed in the experiment. For nonmonitored loans over total loans (xˆ 12 ), equilibrium values in the preshock sessions of two of the three groups are 40% and 79%, which remain close to the average of observed frequencies, 39% and 77%. But for the third group, with a 93% frequency at equilibrium, the average of observed frequencies is only 62%. In the postshock part of the three groups of sessions, theoretical frequencies at equilibrium are 50%, 77%, and 91%, while averages of observed frequencies are at 50.7%, 73% (quite good), and 54.7% (quite bad).14 Therefore, Nash predicted well the frequencies of 14 For x12 (nonmonitored loans), the observed frequencies are farther away from the theoretical frequencies at NE.

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TABLE IV Table of Equilibria Frequencies and Average Frequenciesa Preshock Session 1 2 3 4

Postshock

x11

x12

xˆ 12

x21

x11

x12

xˆ 12

x21

0.08 0.12 0.14 0.10

0.46 0.60 0.55 0.66

0.52 0.68 0.64 0.65

0.69 0.74 0.73 0.63

— 0.21 0.21 0.06

— 0.55 0.36 0.49

— 0.69 0.45 0.50

— 0.76 0.80 0.76

x∗ = 0 093 054 5 5b 6 7 8

0.26 0.09 0.09 0.09 0.07

0.37 0.72 0.71 0.71 0.69

0.49 0.79 0.78 0.78 0.74

x∗ = 0 091 078 0.30 0.49 0.60 0.49 0.51

— — 0.14 0.11 0.24

x∗ = 0 079 022 9 10 11 12

0.22 0.28 0.26 0.26

0.26 0.21 0.35 0.35

0.34 0.29 0.47 0.47

0.08

0.43

0.48

x∗ = 0 05 047

— — 0.81 0.71 0.67

— — 0.54 0.57 0.49

x∗ = 0 077 041 0.60 0.49 0.26 0.26

— 0.18 0.15 0.17

x∗ = 0 04 071 13

— — 0.70 0.63 0.51 — 0.49 0.44 0.43

— 0.50 0.50 0.52

— 0.49 0.44 0.11

x∗ = 0 05 047 0.63

0.14

0.27

0.33

0.67

x∗ = 0 04 071

a

x11 (safe asset), x12 (nonmonitored loans), xˆ 12 (nonmonitored loans with respect to total loans), x21 (effort). A star denotes equilibrium values.

nonmonitored loans in two groups of sessions but not in the third, the one with more extreme equilibrium values. Comparison results are more disappointing for the effort variable x21 . Theoretical equilibrium frequencies of 71%, 22%, and 54% in the preshock sessions are far from the averages of observed frequencies, 40%, 52%, and 70%. Something similar occurs when we compare two postshock theoretical frequencies, 47% and 41%, with the average of observed frequencies, 34.7% and 53%. In the third group of experiments, though, the postshock predicted frequency of 78% coincides with the average of observed frequencies. A natural question to ask is whether the direction of change of frequencies at the mixed Nash equilibrium, as shocks modify payoffs, is a good predictor of the direction of change of the average of observed frequencies. Notice that for nonmonitored loans, the change of the equilibrium frequencies (x∗12 ) after the shock is quite small in two groups of sessions—about 2% only. Even so, in 10 out of 10 cases, the average of nonmonitored loans frequencies moves, after the shock, in the same direction as the theoretical

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´ ez-mart´i bosch-dom` enech and sa

equilibrium frequencies (8 out of 10 for x¯ 12 ). For effort (x21 ), though, only in 6 out of 10 cases the direction of change coincides. For this variable, a one-sided Fisher’s exact test of the null hypothesis of independence of the observed directional variation of x21 and the theoretical directional variation of x∗21 yields a p-value of 0.7. Consequently, we cannot reject the null hypothesis of independence. Next, to gain some additional insights about the accuracy of Nash equilibrium as a predictor of actual frequency changes, we evaluate the association between the size of the change of the theoretical frequencies at the Nash equilibrium and the size of the change of the average of observed frequencies, as shocks occur. The null hypothesis postulates no association between the size of the changes of both variables. For nonmonitored loans, a twosided, two-sample (note that x∗12 only takes two different values, 2% and 25% changes) t-test with equal variances cannot reject the null hypothesis, the p-value being 0.14. For effort, a linear regression between the size of the frequency changes of x∗21 and the size of the average frequency changes of x21 yields a nonsignificant regression coefficient with a p-value of 0.98. This result should not come as a surprise after we could not reject the hypothesis of independence of the observed directional variation of x21 and the theoretical directional variation of x∗21 . Finally, applying the convergence test (6), we observe (see Tables VI and VII in the Appendix) that no session showed tight convergence to (0 x∗12  x∗21 . Session 5, the only one to converge tightly to x∗12  x∗21 , did not converge to x∗11 = 0. In conclusion, whether Nash fails to approximate the average frequencies is a matter of opinion, difficult to settle when we lack an alternative concept of equilibrium to compare with. But the results of the statistical tests performed should be taken as a warning against lightly using, in this context, Nash equilibrium to make predictions about actual behavior. In any case, if equilibrium is understood as a long run outcome of a boundedly rational groping for optimality, the fact that Nash does not predict accurately the average observed behavior might not be a failing of this concept of equilibrium, but the consequence of employing data from sessions that are too short. 4.5. Cycles of Aggregate Behavior The evolutionary model presented in the paper predicts that there exists a threshold x¯ 21 such that, when effort falls below it, a credit contraction takes place.15 But while our general dynamics predicts credit contractions, 15 In the Appendix, see figures for the experimental time series and Table VIII with the values of x¯ 21 for the six different payoffs of the experiment.

cycles of aggregate behavior

125

it is too general to predict their magnitude. We turn now to see how this qualitative prediction fares in our experiment. Testing qualitative predictions of a continuous time model with quantitative results from discrete time experiments requires clearing up some ambiguities. We proceed as follows. Let us define two variables, y and z; y takes value 0 in period t, whenever x21 is below the threshold x¯ 21 in period t, and value 1 otherwise. z takes value 0 in period t whenever the total credit available (1 − x11 ) in period t + 1 is smaller than in period t; z takes value 1 whenever credit available in period t + 1 is larger than in period t; and z takes value 2 whenever credit available in period t+1 stays the same. Under this representation, the claim of the theory (allowing only for one time lag) is that whenever y = 0, it follows that z = 0. We can observe in the experiment that, strictly speaking, the theory is not correct since there are many instances in which this relation does not occur. What we will test, then, is a weaker version of the theoretical results, the null hypothesis claiming the independence of the two variables y and z. Applying a chi-square test, we obtain a p-value of 0.002 which allows us to reject the hypothesis of independence of the two variables. We conclude, therefore, that there is strong evidence that the higher frequency of credit contractions observed after an effort level below the threshold is not due to chance variation. On a different track, a casual reading of the experimental results seems to indicate that the value of the threshold x¯ 21 makes a reasonably good predictor of the existence of serious credit contractions—what we may call credit crunches—in the sense that the higher the value of the threshold, the more likely it is to find a credit crunch. To tighten this observation we define a credit crunch as a situation in which credit availability falls to less than 60%, and we carry a logit analysis of the association between the value of the threshold in each of the experimental half sessions and the appearance of a credit crunch. Although the regression estimate of the logistic model is not statistically significant (p-value = 010),16 the sign of the coefficient is positive. If we evaluate the goodness of the regression fit by the proportion of correctly classified cases, we find that the regression would predict correctly in 78% of the cases. To conclude this part, the statistical analysis of the experimental results confirms that, as predicted by the evolutionary model presented in the paper, there is an association between threshold crossing and credit contraction. The statistical analysis also seems to sustain that the probability

16

The lack of significance may be due, after all, to the small sample size.

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´ ez-mart´i bosch-dom` enech and sa

of credit crunches can be predicted with some accuracy by the value of the threshold, as defined by the theoretical model.

5. CONCLUSIONS In this paper we present an asymmetric game to establish how qualitative behavior predicted by an evolutionary dynamics depends on structural parameters. The game, which we choose to interpret as representing a lender–borrowing relation, has two sets of Nash equilibria: an equilibrium described as full credit availability and a set of equilibria described as credit collapse. Both types of equilibria can be reached under payoff monotonic dynamics. The experiment has helped us to discriminate in favor of the full credit equilibrium. The experiment has also confirmed that Nash equilibrium strategies are not reliable predictors of the average behavior in asymmetric games, as pointed out by Friedman (1996), although convergence to full credit is observed in many sessions. Finally, the evolutionary model presented makes very precise predictions about qualitative changes in behavior, what we have interpreted as cycles of credit contractions followed by credit recoveries. The experimental results have confirmed the theoretical association between the existence of these cycles and the value of a particular threshold. In conclusion, the aggregate behavior observed in the experiments seems consistent with the predictions of the evolutionary model. Note that the experimental results presented in the paper correspond to decision-making on the part of novice players: the sessions run for a limited number of periods and the subjects had no previous experience of the game. In follow-up experiments it would be interesting to confirm that the aggregate behavior of experienced subjects is also consistent with the evolutionary model.

APPENDIX A.1. Proofs Proof of Proposition 1. All states in C are not asymptotically stable under payoff-monotonic dynamics since C is a nonsingleton component of Nash equilibria (Weibull, 1995, Proposition 5.12). The state 1 0 x¯ 21  ∈ C is not Lyapunov stable since x˙ 11 < 0 at all 1 − ( γ( x¯ 21 + δ for all ( δ > 0 and γ ≥ 0 and x˙ 21 > 0 (x˙ 21 < 0) for γ/1 − ( < x∗12 (γ/1 − ( > x∗12 ). Q.E.D.

cycles of aggregate behavior

127

Proof of Proposition 2. For any monotonous dynamics x˙ 11 > 0 (x˙ 11 < 0) at all x = ( x12  x21  with x21 ≤ x¯ 21 (x21 ≥ x21  and ( ∈ 0 1 since by (c.1) and (c.2) SA is the most (least) profitable strategy for all x21 < x¯ 21 (x21 > x21 ). Q.E.D. Proof to Proposition 3. We define the following subsets of the state space  G1 = x ∈ int x12 < 1 − x11 x∗12  G2 = x ∈ int x12 > 1 − x11 x∗12 

(7)

i = x ∈ Gi  x21 = x¯ 21  H

i = 1 2

(8)

Gi1 = x ∈ Gi x21 < x¯ 21 

i = 1 2

(9)

Gi2 = x ∈ Gi  x¯ 21 < x21 ≤ x21  Gi3 = x ∈

Gi  x21

≤ x21 ≤

x∗21 

Gi4 = x ∈ Gi  x21 ≥ x∗21 

i = 1 2

(10)

i = 1 2

(11)

i = 1 2

(12)

Effort (x21 ) is increasing (decreasing) in all x in G1 (G2 ), since the proportion of nonmonitored loans over total loans is smaller (larger) than the equilibrium value x∗21 . Safe asset is the best (worst) strategy in all x in G11 and G21 (G13 ∪ G14 and G23 ∪ G24 ). Monitoring is the best strategy in G12 ∪ G13 and G22 ∪ G23  and the second best in all other states in int. Now we can determine the signs for the rates of growth under any payoff monotonic dynamics (See Table V) and see that the orbits of the dynamical system under payoff monotonic dynamics connect the subsets above as shown in Fig. 7. We prove the following lemma. 1 there exists an x0 ∈ H 2 such that if xt1  = x1  Lemma 1. For all x1 ∈ H 0 then xt0  = x for some t0 < t1 and x˙ 11 t > 0 for all t ∈ t0  t1 . TABLE V Rates of Growth x˙ 11 G11 G12 G13 G14 G24 G23 G22 G21

+ − − − − +

 x˙  12 1 − x11

− − − + + − − −

x˙ 21 + + + + − − − −

FIG. 7. Payoff-monotonic dynamics.

1 is not an absorbing set since x˙ 21 > 0 for Proof. Observe first that H 1 can only be reached from G11 , G11 from  all x ∈ H1 . Figure 7 shows that H  G21 , and the latter from H2 and x11 is increasing at all x ∈ int such that x21 ≤ xˆ 21 . Q.E.D. 1 the orbit either converges to F (x11 is From Fig. 7 we see that from H always decreasing), exhibits a new cycle or converges to C Under either of these possibilities the safe asset exhibits a cycle, since first it increases (from 2 to H 1  and starts decreasing somewhere in G12 to continue decreasing H in G13  G14 , and G24 . Q.E.D. A.2. Payoff Matrices Numbers are in lab units, which will be converted, after each session, at a fixed exchange rate, into pesetas.

FIG. 8. Payoff matrices.

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129

A.3. Convergence to x∗12 and x∗21 Convergence is shown in Tables VI and VII. TABLE VI Convergence toward Mixed Equilibrium (xˆ 12 = x∗12 a Session 1 2 3 4 5 6 7 8 9 10 11 12 13

Periods

Preshock

Periods

1–32 1–16 1–16 1–25 1–52 1–19 1–20 1–22 1–30 1–16 1–16 1–25 1–16

L(32) T(15), L(12) No convergence T(25), L(15) T(52), L(49) No convergence No convergence No convergence T(30) T(16), L(15) No convergence L(1) No convergence

— 17–32 17–32 26–62 — 20–50 21–50 23–50 — 17–34 17–34 26–62 17–34

Postshock — No convergence No convergence No convergence — T(50), L(20) No convergence T(23) — L(34) No convergence T(28), L(26) T(34), L(32)

a

T(t) and L(t) stand respectively for tight and loose convergence from period t onward. Of course, T(t) implies L(t).

TABLE VII Convergence toward Mixed Equilibrium (x21 = x∗21 a Session 1 2 3 4 5 6 7 8 9 10 11 12 13 a

Periods

Preshock

Periods

1–32 1–16 1–16 1–25 1–52 1–19 1–20 1–22 1–30 1–16 1–16 1–25 1–16

L(1) No convergence No convergence L(1) T(49), L(1) No convergence L(19) L(1) L(19) No convergence T(16) T(25), L(23) No convergence

— 17–32 17–32 26–62 — 20–50 21–50 23–50 — 17–34 17–34 26–62 17–34

Postshock — T(20), L(17) No convergence L(26) — L(45) T(16), L(21) L(23) — No convergence T(32), L(25) No convergence No convergence

T(t) and L(t) stand respectively for tight and loose convergence from period t onward. Of course, T(t) implies L(t).

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´ ez-mart´i bosch-dom` enech and sa A.4. Values of the Threshold x¯ 21

Threshold values are given in Table VIII.

TABLE VIII Values of the Threshold x¯ 21 Payoffs

x¯ 21

P1 P2 P3 P4 P5 P6

0.24 0.46 0.15 0.36 0.40 0.23

A.5. Dynamics of Credit and Effort In Figs. 9–21 we plot the dynamics for total credit (

), effort (, ,), equilibrium values (in the mixed equilibrium) of total credit (— · — · —) and effort (— — — —), and the critical threshold x¯ 21 (— · · — · ·).

FIG. 9. Session 1.

cycles of aggregate behavior

FIG. 10. Session 2.

FIG. 11. Session 3.

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´ ez-mart´i bosch-dom` enech and sa

FIG. 12. Session 4.

FIG. 13. Session 5.

cycles of aggregate behavior

FIG. 14. Session 6.

FIG. 15. Session 7.

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´ ez-mart´i bosch-dom` enech and sa

FIG. 16. Session 8.

FIG. 17. Session 9.

cycles of aggregate behavior

FIG. 18. Session 10.

FIG. 19. Session 11.

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FIG. 20. Session 12.

FIG. 21. Session 13.

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