The effect of endogenous timing on coordination under asymmetric information: An experimental study

Games and Economic Behavior 86 (2014) 264–281

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Games and Economic Behavior www.elsevier.com/locate/geb

The effect of endogenous timing on coordination under asymmetric information: An experimental study ✩ Francesco Brindisi a , Bo˘gaçhan Çelen b , Kyle Hyndman c,∗ a b c

NYC Office of Management and Budget, 255 Greenwich Street, New York, NY 10007, USA Melbourne Business School, 200 Leicester Street, Carlton, VIC 3053, Australia Naveen Jindal School of Management, University of Texas at Dallas, 800 W. Campbell Rd (SM31), Richardson, TX 75080, USA

a r t i c l e

i n f o

Article history: Received 10 August 2011 Available online 13 April 2014 JEL classification: C72 C91 D62 D82 D83 Keywords: Coordination Delay Information Global games Experiment

a b s t r a c t This paper investigates the role of endogenous timing of decisions on coordination under asymmetric information. In the equilibrium of a global coordination game, where players choose the timing of their decision, a player who has sufficiently high beliefs about the state of the economy undertakes an investment without delay. This decision (potentially) triggers an investment by the other player whose beliefs would have led to inaction otherwise. Endogenous timing has two distinct effects on coordination: a learning effect (early decisions reveal information) and a complementarity effect (early decisions eliminate strategic uncertainty for late movers). The experiments that we conduct to test these theoretical results show that the learning effect of timing has more impact on the subjects’ behavior than the complementarity effect. We also observe that subjects’ welfare improves significantly under endogenous timing. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Many economic activities are characterized by strategic complementarities where individuals achieve desirable outcomes if they can coordinate their actions. Such activities include the agglomeration of businesses (Caplin and Leahy, 1998), technology adoption (Katz and Shapiro, 1985, 1986), crop choice by subsistence farmers (Conley and Udry, 2001), bank runs and currency attacks (Diamond and Dybvig, 1983; Morris and Shin, 1998 and Goldstein and Pauzner, 2005), and foreign direct investment (FDI) (Jordaan, 2009), among others.

✩ We would like to thank the editor, Thomas Palfrey, an advisory editor and two anonymous referees for their valuable comments, which have substantially improved the paper. We gratefully acknowledge helpful discussions with Alessandro Citanna, John Duffy, Stephen Morris, Hyun Song Shin, and Paolo Siconolfi, as well as seminar participants at New York University, University of Maryland, Rutgers University, Columbia University, Fordham University, University of Texas at Austin, Boston College, Carnegie Mellon University, University of British Columbia, McGill University, Yeshiva University and Georgetown University. This paper has also benefited from suggestions by the participants of the AEA 2009 Annual Meeting, and the 2009 ESA North American Conference. The views expressed in this paper are the personal views of the authors and are not necessarily reflective of views of the City of New York or its Office of Management and Budget. Corresponding author. E-mail addresses: [email protected] (F. Brindisi), [email protected] (B. Çelen), [email protected] (K. Hyndman). URLs: http://bogachancelen.com/ (B. Çelen), http://www.hyndman-honhon.com/Main/Kyle_Hyndman.html (K. Hyndman).

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http://dx.doi.org/10.1016/j.geb.2014.03.018 0899-8256/© 2014 Elsevier Inc. All rights reserved.

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Another common characteristic of the examples mentioned above is the existence of asymmetric information. For instance, potential investors in foreign markets have myriad sources of information regarding the uncertainty that their decision involves. Uncertainty in FDI decisions can include political uncertainty (Rodrik, 1991), demand uncertainty (Goldberg and Kolstad, 1995), and cost uncertainty (Creane and Miyagiwa, 2007). In many cases, individuals acquire private information from various sources regarding these uncertain events. For example, as Markusen (2004) argues, “[... a] multinational firm may adopt some contractual agreement with a local agent as a means of exploiting any superior information the agent may possess regarding market characteristics.” As a result, each firm’s information is determined by its relationship with a local agent, which can create asymmetry because different investors pair with different local agents leading to different sources of information. The effect of strategic timing has been subject to analysis both under pure information externalities (Chamley and Gale, 1994; Gul and Lundholm, 1995) and under strategic complementarities (Bolton and Farrell, 1990; Farrell, 1987; Farrell and Saloner, 1985, 1988). Under pure information externalities, time serves as a medium for disseminating information between individuals who make their decisions at different moments. We will refer to this effect of time as the learning effect. In the presence of strategic complementarities, time serves as a coordination device, because early movers eliminate strategic uncertainty by late movers, thus facilitating coordination. In the sequel, we will refer to this effect as the complementarity effect of strategic timing. As will be seen, the two effects can be distinguished by comparing decisions in games with and without uncertainty. Given no uncertainty there is nothing to be learned, so all differences between the simultaneous and sequential treatments are attributable to complementarity effects. With uncertainty, treatment effects are a combination of complementarity and learning effects. The outcome of economic activities can also be affected by the timing of decisions. A firm’s investment decision reveals valuable information about the profitability of the project. Early investors therefore trigger a process through which information aggregates. At the same time, the option of having access to this information in later periods (i.e., the learning effect) creates an incentive for the investors to wait and observe others’ decisions. In fact, as Chamley and Gale (1994) show, if the incentive to delay the investment is sufficiently strong, no investment is undertaken even though it is beneficial for all investors. Despite the substantial amount of research analyzing the complementarity and learning effects separately, there is little theoretical work (viz. Brindisi et al., 2013; Dasgupta, 2007; Xue, 2003) that studies the two effects together. Furthermore, to the best of our knowledge, there is no experimental research that does this. Little is known about the interaction of these two effects. We aim to fill this gap by using a series of laboratory experiments to investigate the effects of strategic timing decisions on coordination under asymmetric information—i.e., when strategic complementarities and information externalities coexist. Since the main focus of our paper is on strategic complementarities and asymmetric information about the fundamentals of the economy, we base the design of our experiment on the global coordination game introduced by Carlsson and van Damme (1993).1 Incorporating endogenous timing into this model allows us to analyze the two effects of timing, which we discussed above. The theoretical framework analyzed by Brindisi et al. (2013) provides us with insights and hypotheses that we test in our experimental treatment. They characterize the equilibrium of a simple two-player global coordination game that allows endogenous timing. They demonstrate that the complementarity and the learning effects of timing allow players to internalize the returns from coordination, and strategic delay serves as a coordination device. In particular, they find that an optimistic player invests earlier, and a pessimistic player invests later, if at all. The intuition for this is as follows. An optimistic investor, already expects a high return, making investment an attractive option. Beyond this, the optimistic investor also understands that his investment will make the other player more optimistic about the economic fundamentals, increasing the likelihood that the other player will invest in subsequent periods, if he has not already done so. Because of strategic complementarities, this further increases the optimistic investor’s expectation of higher returns. The equilibrium behavior of the pessimistic investor is as follows: Because of his pessimistic beliefs, he not only expects a low return, but he also believes that the other investor holds pessimistic beliefs as well. Therefore, he finds it optimal to wait. If he observes investment, then in the next period he becomes less pessimistic, and adjusts his decision accordingly. This characterization determines the effect of endogenous timing on welfare. It is well-known that the risk-dominant equilibrium is the limit equilibrium of the simultaneous game as the player’s signal becomes fully informative. However, this equilibrium is inefficient. In contrast, Brindisi et al. (2013) show that the equilibrium of the sequential game is fully efficient in the limit as signals become fully informative. For the parameters used in our experiment, the ex ante average payoffs are higher in the sequential game than in the simultaneous game for all levels of informativeness of signals. However, the difference between the two treatments is actually decreasing for all levels of signal informativeness. Thus, endogenous timing is especially beneficial in relatively noisy environments.

1

For a thorough review of global games literature, see Morris (2008) and Morris and Shin (2003).

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As noted, Dasgupta (2007) and Xue (2003) study similar questions with a theoretical approach.2 Dasgupta (2007) provides an analysis of a two-period game with endogenous timing and a continuum of players. In the second period of the game, players who still have an option to invest observe an additional private signal that provides information about the aggregate level of investment in the first period. The second period signal in Dasgupta (2007) differs from the information conveyed in our model by an investment decision in two respects: First, it is privately observed, and second, it is subject to idiosyncratic noise. This modeling choice has implications in terms of the interplay between information and timing, which is the main focus of the present paper. In Dasgupta (2007), players who invest in the first period know that they collectively play a role in determining second-period signals. Therefore, they take into account the aggregate influence on investment decisions in the second period, yet, because of the continuum of players, the individual signaling effect is negligible. Under specific distributional assumptions, Dasgupta (2007) shows that if private information is sufficiently precise, the game has a unique equilibrium in monotone strategies. However, because the signaling effect is negligible the equilibrium does not achieve efficiency at the complete information limit. Xue (2003) also studies a dynamic coordination game under incomplete information. However, Xue (2003) assumes that player types are not correlated. The correlation between the types (namely, player’s private information regarding the common return of investment) is a critical feature of the current paper because it allows players to signal about the value of investment, when they choose to invest in early periods. Xue (2003) also analyzes the role of irreversibility of actions in dynamic coordination games. Similarly, Dasgupta et al. (2012), and Kováˇc and Steiner (2013) focus on the role of irreversibility in dynamic global coordination games. However, they assume that players do not observe each other’s actions. Hence, they do not focus on social learning dynamics as we do in the present paper.3 The laboratory setting provides us with an environment in which we can observe empirical regularities while we control for asymmetric information—a variable that is challenging to observe or identify in market data. We base the experiment on a specific parameterization and vary the informativeness of private information in both simultaneous and sequential games to obtain a rich data set. The analysis of the data shows that the subjects’ actual behavior is consistent with many properties of equilibrium behavior. For example, although there are three periods in which the subjects can make their decision, there are negligibly few subjects who delay the investment to the third period. Instead, if a subject has optimistic beliefs, he invests in the first period; if his beliefs are more pessimistic, he waits until the second period to invest (if at all). We also see that a subject who delays his decision invests in the second period, only if he observes that the subject invests, and if he is not too pessimistic. Otherwise, he never invests. Another main finding of our experiment is that endogenous timing is a significant instrument for enhancing welfare, except when there is no uncertainty. When subjects have complete information, the coordination rates between simultaneous and sequential games are nearly identical. This suggests that the complementarity effect of timing is not a strong determinant of the outcome. When there is uncertainty, we find that coordination about the efficient outcome is uniformly higher in the sequential game than in the simultaneous game. Consistent with our theoretical findings, the gap between the coordination rates of simultaneous and sequential games increases when private information becomes less informative. That is, the more information asymmetry there is between the players, the more endogenous timing facilitates coordination. We confirm the same results when making similar comparisons in terms of subjects’ actual earnings. Overall, we conclude that in the lab the learning effect of timing is stronger than the complementarity effect of timing. Like the theoretical literature on global games, the experimental literature can be categorized as dealing with static or dynamic global games. The pioneering work on static global games is Heinemann et al. (2004). They present three main findings: First, differences in observed behavior under complete and incomplete information are not significant.4 Second, behavior is consistent with the comparative statics of global games. Third, subjects cooperate more often than the predictions of the theory. Duffy and Ochs (2012) is the bridge between static and dynamic global games experiments. In a similar environment to the complete information treatments of Heinemann et al. (2004), they study behavior in both static and dynamic games.5 Contrary to their focus on complete information, our experiments examine behavior in both static and dynamic games under incomplete information, allowing us to provide a clear test of the theory of dynamic global games that we consider. In its static treatments, Duffy and Ochs (2012) elicit subjects’ strategies and find that actual decisions are often consistent with the implied, elicited cutoff strategies, though they also find that there is substantial variance in individual cutoffs and that cutoffs vary with changes in the environment in ways not predicted by the theory. Comparing the dynamic and static

2 Angeletos and Werning (2006), Angeletos et al. (2007) and Heidhues and Melissas (2006) also share some similarities. Among other results, the first paper shows that the endogenous transmission of information can lead to a multiplicity of equilibria, while the second shows in a dynamic model of regime change the interplay of endogenous and exogenous learning about the economy also generates a multiplicity of equilibria. Finally, Heidhues and Melissas (2006) examine the role of cohort effects, and characterize the conditions under which uniqueness of equilibrium is established. 3 Our paper also shares common elements with the literature on observational learning with pure information externalities. A few papers focus on endogenous timing and hence, strategic delay (cf. Caplin and Leahy, 1994; Chamley and Gale, 1994; Gul and Lundholm, 1995 and Levin and Peck, 2008). In our paper, the additional presence of payoff externalities causes a fundamental change; because now an investor must weigh the benefits of moving early to transmit information against the benefits of waiting to observe others’ decisions. 4 Similarly, Cabrales et al. (2007) test Carlsson and van Damme (1993) and provide further support that complete and incomplete information environments lead to similar behavioral patterns. They show that behavior converges to theoretical predictions as the subjects become more experienced. 5 Fehr and Shurchkov (2008) and Costain et al. (2007) also conduct dynamic global games experiments, though because the underlying models that they study have multiple equilibria, they are separate from our focus.

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i

w

i

θ , θ

θ − 20 , 25

w

25 , θ − 20

25 , 25

267

Fig. 1. Player’s returns for a realization of Θ .

games, Duffy and Ochs (2012) find that cutoffs do not differ in the two treatments. That is, while subjects have lower cutoffs than theory predicts in the static game, they have higher cutoffs than the efficient subgame perfect equilibrium threshold in the dynamic games. Moreover, in the dynamic games, they find that subjects adopt a “wait-and-see” approach, which often leads to substantial delay. Our experimental data share some similarities with these papers. Like Heinemann et al. (2004) and Duffy and Ochs (2012), we find that subjects have lower thresholds than predicted by the global games theory in the static game. Also, like Duffy and Ochs (2012), we find that subjects in our dynamic treatments have higher thresholds than predicted by our theory, and that the thresholds in the dynamic and static treatments are indistinguishable under complete information. However, unlike Duffy and Ochs (2012), our experiment involves incomplete information. Under incomplete information, we find that behavior is distinguishable—in particular, thresholds are lower and the frequency of coordination is higher in the dynamic games than in the static games. Finally, in a related strain of literature, Sgroi (2003), Ivanov et al. (2009), and Ivanov et al. (2013) study endogenous timing in social learning games. These experiments focus only on the role of timing decisions in information acquisition and investigate various behavioral phenomena that arise in social-learning environments with pure informational externalities. In contrast, our game has both payoff and information externalities. We focus on the interplay between these externalities as well as their effect on welfare. The rest of the paper is organized as follows: Section 2 outlines the main predictions of the theoretical model, parameterized as in our subsequent experiment. From this, we develop a number of hypotheses, which we subsequently test. Section 3 explains the experimental design and Sections 4 and 5 provide an analysis of the data. In particular, in Section 4, we examine only choice behavior, while in Section 5, we use elicited beliefs to gain further insights into subjects’ decision making processes. Finally, Section 6 concludes. 2. Theory In this section, we provide the theoretical framework, which generates the hypotheses that will be subsequently tested in our experiment. In the interest of brevity and simplicity, we directly present the model as implemented in the lab. A more general treatment, along with formal proofs, can be found in our companion work (Brindisi et al., 2013). 2.1. Preliminaries The game is based on the global coordination game of Carlsson and van Damme (1993). Two players, i = 1, 2, make a binary investment decision. There are two actions available to them: i and w. We interpret action i as investing, and w as waiting. While w is a safe action, whose payoff is normalized to 25, the return of investment is determined by a random variable Θ . Given a realization θ , the payoffs are as in Fig. 1. In particular, when both players take action i the return is θ , but if only one player takes action i his payoff is θ − 20. That is, the actions exhibit strategic complementarities: If a player invests, the return of investment is higher by an amount 20 when the other player invests as well. The random variable Θ is uniformly distribution over [20, 50]. The true state, θ , is realized prior to investment decisions. Following the realization of θ , but before making any decision, each player receives a private signal. Player i’s signal is determined by the random variable Xi defined as

Xi := Θ + Ei , where Ei is a uniform distribution over [−e , e ]. In the experiment, we run treatments where e takes values 0, 2, 5, or 10. We assume that E1 and E2 are identically and independently distributed, and that Ei and Θ are independent for i = 1, 2. We assume that players are Bayesian. Therefore, when player i receives a signal xi he updates his belief about Θ by Bayes’ rule. Player i’s posterior belief about Θ , conditional on observing xi , is represented by the distribution function F Θ (·|xi ). Since both Θ and Ei are distributed uniformly, F Θ (·|xi ) is a uniform distribution with support [max{20, xi − e }, min{50, xi + e }]. A player also forms beliefs about the other player’s signal. Player i, conditional on observing xi , updates his belief about the signal of the other player, which we denote by F X j (·|xi ). For interior cases of xi ∈ [20 + e , 50 − e ], the density f X j (·|xi ) is a symmetric triangle, centered at the signal xi , the reader can refer to an online appendix for complete computations.6

6

The appendix is available at http://bogachancelen.com/research.html.

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Finally, observe that for xi > 45 + e, it is a strictly dominant strategy for player i to invest, while for xi < 25 − e, it is a strictly dominant strategy to never invest. 2.2. The game The game consists of τ periods: t = 1, 2, . . . , τ . In each period, each player i takes an action ati ∈ {i, w}. The action i is irreversible, while w is reversible. Therefore, if player i takes action i in period t (ati = i), then his action is determined to be i in all future periods. If player i’s action was w up to period t, player i can still choose between actions i and w in period t + 1. However, switching to action i is costly in periods t > 1: If a player switches from action w to i in period t, then he pays a cost of 2(t − 1); that is, the cost of delayed investment is c = 2 per period of delay. For a given θ , payoffs are determined by the sequence of actions taken in τ periods:

ui





τ at1 , at2 t =1





=

25

if aτi = w,

θ − 20 1aτj =w − 2(t − 1) if ati −1 = w and, ati = i,

where at = (at1 , at2 ) is the profile of actions taken in period t, and 1aτ =w is an indicator function that takes value 1 if j aτj = w and 0 otherwise. We assume that players have perfect information about the history and the solution concept is perfect Bayesian equilibrium. A player’s strategy determines his action at each history given his signal. As it is standard in the literature, in our equilibrium analysis, we will focus on monotone strategies: If a player invests for a signal xi , then he also invests for signals greater than xi . Thus, a monotone strategy is characterized by thresholds, one for each history. We denote the threshold of player i in period t by khi . Under a monotone strategy, after a history h, a player i, invests in period t if xi ≥ khi ; otherwise, he waits. 2.2.1. Simultaneous game (τ = 1) If τ = 1, the game reduces to a simultaneous game (henceforth sim), which is analyzed extensively in the literature of global games. For completeness, we will briefly review the key aspects of the game. Since there is only one period, a single threshold characterizes a player’s monotone strategy. So, we will write κi to denote player i’s threshold. Note that, if player j follows a monotone strategy with threshold κ j , then player i, who received signal xi , expects him to invest with probability 1 − F X j (κ j |xi ), and to wait with probability F X j (κ j |xi ). Hence, player i’s expected payoff from action i is E[Θ|xi ] − 20F X j (κ j |xi ). Consequently, player i’s best-response is to invest if and only if his expected payoff from investment is at least as much as the safe payoff 25. If we restrict our attention to symmetric equilibria in monotone strategies, an equilibrium with threshold κ := κ1 = κ2 must satisfy the indifference condition E[Θ|κ ] − 20F X j (κ |κ ) = 25. Given the parameterization of our model, this indifference condition becomes κ − 20(1/2) = 25, which means that κ = 35. In fact, it is well-known that this is the unique equilibrium. We state this result in the following proposition. Proposition 1. There exists a unique equilibrium of the sim game. The equilibrium is in monotone strategies and symmetric. It is characterized by the threshold κ = 35. 2.2.2. Sequential game (τ ≥ 2) Players’ strategic considerations change dramatically when the game becomes sequential (henceforth seq). In the seq game, on one hand a player can wait in order to benefit from the information that can be obtained from the other player’s investment decision; on the other hand, he can invest in the current period, avoid the cost of investment in the future, and more importantly, reveal information to the other player. This information may convince the other player to invest, allowing both players to benefit from the complementarities. We begin our analysis with two preliminary observations. First, observe that if the cost of delay is so large that a player never finds it profitable to invest after the first period, then the sequential game essentially collapses to a simultaneous game. In our parameterization of the game for our experiment, we keep the cost of delay, c = 2, small enough so that the distinction between the seq and sim games is meaningful. Second, observe that in any monotone-strategy equilibrium, once both players wait, there cannot be any investment in the future. The intuition behind this result is simple. At any point in time, a player decides whether or not to invest by comparing the expected payoff of investing, and the option value of waiting. In an equilibrium with monotone strategies, the observation that the other player waits reveals bad news. This, in turn, makes the player more pessimistic, and hence, lowers the expected payoff of investing. As a result, a player who has not already invested, finds it even less profitable to invest after observing w. Since this argument applies to both of the players, there is never any investment after period two. The above observations greatly simplify our equilibrium analysis because they imply that in seq investment will only occur in the first two periods, if at all. Therefore, in any monotone-strategy equilibrium, player i has two thresholds k1i and k2i for periods 1 and 2, respectively. If xi ≥ k1i , then player i invests in the first period. Alternatively, if k2i ≤ xi < k1i , then player i will invest in the second period, provided that the other player invested in period one. In other words, over the range

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of signals [k2i , k1i ], player i’s signal is not sufficiently strong to justify immediate investment, but he will invest in period two if he observes an investment. This is so because (1) observing the other player invest makes him more confident about the state, since he infers that x j ≥ k1j , and (2) it eliminates strategic uncertainty. Finally, if xi < k2i , then his signal is not high enough to justify a delayed investment, even if he observes an investment; so player i never invests. We state the equilibrium characterization for our experiment in the next proposition. A more general result and formal proof is given in Brindisi et al. (2013). Proposition 2. There exists a unique equilibrium of the seq game in monotone strategies. The equilibrium is symmetric (i.e., kt1 = kt2 = kt , for t = 1, 2) and characterized as follows:

⎧ ⎨i

in the first period if xi ≥ k1 , Play i in the second period if a1j = i and xi ≥ k2 , ⎩ w otherwise. The equilibrium value of k2 is simply the signal for which a player is indifferent between the safe return (25) and the expected return from investment (θ ) minus the cost of delayed investment, given his signal xi = k2 and the observation that the other player invested. Each player’s problem in the first period is whether to invest right away, or to wait and decide in the second period at a cost c = 2. Investing in the first period involves the risk of incurring the cost of miscoordination (20) in case the other player never invests (i.e. his signal is below k2 ), as well as the benefit of convincing the other player to invest if he waits to observe an investment in order to invest (i.e. his signal is between k2 and k1 ). Waiting, on the other hand, has the benefit of learning more about the return of investment and eliminating the strategic uncertainty, but it is costly. The signal, for which a player is indifferent between these two options, given the equilibrium value of k2 , determines k1 . Table 1(a) reports the equilibrium values of k1 and k2 for the parameters of the experiment. 2.3. Discussion of equilibrium predictions In this section, we will review the equilibrium predictions of the sim and seq games, and will develop our hypotheses. Recall that, in the experiment, it is a dominant strategy to take some actions for some range of signals. In particular, for xi > 45 + e, it is a strictly dominant strategy for player i to invest, while for xi < 25 − e, it is a strictly dominant strategy to never invest. Thus, as a straightforward test of basic rationality, we expect that subjects should never play dominated strategies. There are two more theoretical predictions, that can be viewed as fundamental tests of subjects’ understanding of the problem. First, in the seq game, subjects can intuit that observing the other player not invest reveals bad news. Therefore, subjects who initially wait, should invest in the second period, only if they observe that the other player invested. If this turns out to be true, then we should observe no investment in period three. We collect these predictions in the following hypothesis: Hypothesis 1. (a) In both the sim and seq treatments, subjects do not play dominated strategies. (b) In the seq game, subjects invest in period 2 only if they observe their match invest in period 1. (c) In the seq game, all investment takes place in period one or period two. The equilibrium analysis is based on the assumption that players use monotone strategies. In the sim game, for e > 0, there is a unique equilibrium in monotone strategies; in the seq game we assume that the strategies are monotone and characterize the equilibrium. This leads to the obvious question of whether subjects’ behavior is consistent with the use of monotone strategies. This is important for at least two reasons. First, it can help us understand whether the equilibrium provides a meaningful benchmark to analyze the data. Second, we can analyze the data under the light of theoretical comparative statics, and if there are systemic deviations, we can investigate when and how subjects’ behavior cause these deviations. Moreover, previous studies (e.g., Heinemann et al., 2004) suggest that subjects do, in fact, use monotone strategies. Hypothesis 2. In both the sim and seq treatments, subjects use monotone strategies. The equilibrium analysis of the seq and sim games sheds light on the learning and complementarity effects of timing. Let us first discuss the benchmark cases of sim and seq under complete information. We assume that θ > 25 + c = 27, so that investing in the second period remains a feasible option. The equilibria of seq and sim under complete information are straightforward. While there are multiple equilibria in the sim game, the seq game has a unique subgame perfect equilibrium in which both players invest in the first period if and only if θ ≥ 27. In the sim game, the reason for multiple equilibria is self-fulfilling beliefs: If both players believe that the other will play i (w) then it is optimal to play i (w). On the other hand, the uniqueness of the subgame perfect equilibrium in the seq game depends on players’ beliefs that if they invest in the first period for θ ≥ 27, then the other player—even if he has not already invested—will invest in the second period. So, under complete information, the possibility that the players

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can choose the timing of their decisions has an important impact on the equilibrium prediction of the game through the complementarity effect of timing.7 When there is uncertainty and asymmetric information about the fundamentals, the equilibrium prediction changes dramatically in the sim game. In the Bayesian equilibrium of the game, when both players have a signal above the equilibrium threshold κ = 35, they coordinate on the outcome (i, i). Similarly, they coordinate on the outcome (w, w) when both signals are below 35. Finally, the players fail to coordinate when one signal is above, and the other one is below 35. The literature on global games shows that these failures vanish as the players’ signals become fully informative. In fact, in the limit, there exists a unique equilibrium. Given our parameterization, this equilibrium is characterized by κ = 35. An immediate consequence of this result is that even though the efficient outcome is to invest when the expectation of θ exceeds the safe return 25, this does not necessarily occur in the limit equilibrium. In the case of seq, under complete information, there is a unique subgame perfect equilibrium because a player knows that his investment in the first period guarantees coordination. Under asymmetric information, the explanation behind the perfect Bayesian equilibrium is similar. Precisely, if a player invests in the first period, the probability that the other player will invest in the second period increases. The investment in the first period affects the probability of coordination in two ways: First, as in the complete information case, there is no risk of miscoordination for a player who did not invest in the first period. Second, since a player invests in the first period only if his signal is high enough (above k1 ), observing investment makes other player’s beliefs more optimistic about the overall profitability of investment. As a result, the equilibrium exhibits sorting; the players whose signals are high enough move first, while those who do not invest in the first period benefit from the information revealed from the investment decision. The sequential structure of seq makes it possible to exchange information (learning effect), and use timing as a coordination device (complementarity effect). Note that in the limit case of seq when the private signals become increasingly informative and the cost of delay becomes vanishingly small, the unique limiting equilibrium is the subgame perfect equilibrium of the seq game under complete information. In other words, there is no discontinuity in the limit of the seq game, and full efficiency is achieved (i.e., both players invest if and only if θ ≥ 25). Table 1 reports several properties of the equilibria for the parameters used in our experiment in both the sim and seq games. Table 1(a) summarizes theoretical thresholds at different noise levels. We say that a pair of subjects coordinate on investment, if they both invest in a round. If however, only one subject invests, we say that they miscoordinate. Table 1(b) reports the frequencies of coordination on joint investment, (i, i), and miscoordination, (i, w), if subjects play according to the theoretical predictions for both the seq and sim games. The comparison of sim and seq games lead to Hypothesis 3. First, because players have an additional period to invest in the seq game, they are more likely to invest for lower signals in the seq game; hence period one thresholds should be lower. Second, because observing investment reveals good news about the state, subjects with relatively low signals will invest in period two. Hence, the sorting property of equilibrium behavior in the seq game means that coordination on investment should be higher and miscoordination should be lower in the seq treatment. Indeed, Table 1(a, b) shows that these differences can be quite large; e.g. coordination on joint investment in the seq game is predicted to be 53% to 87% higher than in the sim game. Hypothesis 3. In all treatments, (a) k1 ≤ κ , (b) coordination on investment in the seq game is higher than in the sim game, (c) miscoordination in the seq game is lower than in the sim game. Because of the greater probability of coordination on investment, and the lower probability of miscoordinating in the seq treatments, it is not surprising to find that subjects have a higher expected payoff in the seq treatments. This is what Table 1(c) reports. What is surprising is that the difference between expected payoffs in the two treatments is actually increasing in the noisiness of signals. The main driver for this result can be seen by looking at the coordination and miscoordination rates in Table 1(b). In the sim game, as noise increases, the coordination rate rapidly declines and the miscoordination rapidly declines. In contrast, the change in coordination and miscoordination rates is significantly moderated in the seq treatment. Thus, while endogenous timing is uniformly welfare improving, it is most beneficial in noisy environments. Although we could state our hypothesis in terms of expected payoffs, because the distributions of signals vary across treatments, we seek a metric that is partially robust against randomness in signals and states of the world. The particular metric we adopt is the payoff ratio, π / max{θ, 25}. That is, we take the payoff that a subject actually receives and normalize it by the maximum possible payoff he could receive (i.e., in the best equilibrium of the complete information setting). Table 1(d) shows the theoretical predictions for the payoff ratio. Similar to the expected payoff, subjects have a higher

7 Note that if θ < 27, the seq game essentially becomes a sim game since no player will invest in the second period. Therefore, for θ ∈ [25, 27) equilibria of seq and sim coincide.

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271

Table 1 Theoretical predictions for experimental parameters. (a) Equilibrium Thresholds Noise

0 2 5 10

sim

seq

κ

k1

k2

[25, 45]a

[25, 27]b

35 35 35

27.79 28.92 30.74

27 25.61 23.54 20.13

(b) Coordination and Miscoordination Rates Noise

0 2 5 10

Coordination on investment

Miscoordination

sim

seq

% difference

sim

seq

% difference

0.500 0.478 0.445 0.389

0.767 0.759 0.747 0.730

53.38 58.68 68.07 87.74

0 0.044 0.111 0.222

0 0.004 0.011 0.023

–

−90.56 −90.11 −89.66

(c) Expected Payoff Noise

sim

seq

% difference

0 2 5 10

33.755 33.288 32.504 30.975

35.355 35.245 35.028 34.521

4.74 5.88 7.76 11.45

(d) Payoff Ratio

π / max{25, q}

Noise

sim

seq

% difference

0 2 5 10

0.947 0.934 0.912 0.871

0.998 0.994 0.986 0.969

5.32 6.38 8.07 11.30

a Note that the limit equilibrium as e → 0 is for a threshold of 35 in the sim game. Predictions in panels (b–d) are based on this limit equilibrium. b Note that the limit equilibrium as e → 0 is for a threshold of 27 in the seq game. Predictions in panels (b–d) are based on this limit equilibrium.

payoff ratio in the seq treatment, and the difference between seq and sim is increasing in the noisiness of signals. We summarize these predictions as follows. Hypothesis 4. In all treatments, (a) the payoff ratio in the seq game is higher than in the sim game, (b) the difference of payoff ratios between the seq game and the sim game is increasing in the noisiness of signals. 3. Experimental design The experiment was run at the Center for Experimental Social Sciences (C.E.S.S.) of New York University. 308 subjects were recruited from undergraduate classes at NYU, who had no previous experience in our experiments. In each session, after the subjects read the instructions, they were also read aloud by an experimental administrator.8 Each session lasted for about 90 minutes, and each subject participated in only one session. An $8 participation fee, and subsequent earnings, which averaged about $21, were paid in private at the end of the session. Throughout the experiment, we ensured anonymity and effective isolation of subjects in order to minimize any interpersonal interactions and influences that could stimulate a specific pattern of behavior.9 The experiment was programmed in z-Tree (Fishbacher, 2007). Appendix A contains the instructions for one of the sessions. We report two treatments: sim and seq. For each treatment, we ran different sessions in which we varied the informativeness of subjects’ signals. In each session, subjects played the specified game for 40 rounds. Subjects were randomly matched with another subject in the laboratory at the beginning of each round.

8 At the end of the first round, the subjects were asked if there were any questions. No subject reported any problems with understanding the procedures or using the software. 9 Participants’ workstations were in isolated cubicles, making it impossible for them to observe others’ screens or to communicate. We also made sure that all the participants remained silent throughout the session. At the end of a session, participants were paid in private according to the number on their workstation.

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Table 2 Summary of treatments. Treatment

Number of subjects

c

e

20 – 20 20

2 2 2 2

0 2 5 10

20 – 16 14

na na na na

0 2 5 10

Regular

Beliefs

seq seq seq seq

22 24 28 32

sim sim sim sim

16 16 24 36

We also ran a variant of sim and seq treatments, in which we elicited subjects’ beliefs about what their match will play, and about the state variable θ .10 We will label those sessions in which we elicited beliefs as Beliefs treatments, otherwise we will call them Regular treatments. Below, we will explain the differences between Beliefs and Regular treatments in more detail. Throughout the experiment we used a neutral language and replaced the terms investing, waiting and signal with action a, action b, and estimate, respectively. Table 2 provides a summary of our design. The parameter e indicates the noisiness of signals. Henceforth, we will write seq(e ) and sim(e ) to refer to the treatments with corresponding parameter e. In all seq treatments the cost of delay, c, was 2. Sequential (seq) In all treatments, subjects played the game analyzed in the previous section, with payoffs (excluding the cost of delayed investment) given in Fig. 1. Recall that the distribution of the state variable Θ was uniform over the support [20, 50] and that conditional on the state, θ , each subject i received an independent signal xi which was drawn from a uniform distribution over the support [θ − e , θ + e ]. The experimental software randomly generated θ , as well as signals x1 and x2 for each pair in every round. In the Regular treatments, the length of the game was set to be τ = 3, while in the follow-up Beliefs treatment, we set τ = 2. In each period, subjects simultaneously chose i or w. Choosing i was irreversible, while choosing w was reversible. If a subject chose w in the first period, then in the second period, he would observe the first-period decision of his match. Similarly, in the third period, if the subject chose w in periods one and two, he would observe his match’s previous decisions. The cost of investment was only incurred if the subject chose i in period two or three, after having previously chosen w. The cost of investment in period two was 2 points, while it was 4 points in period three. Immediate investment was costless. Simultaneous (sim) In the sim treatment the payoffs were identical to those in the seq treatment. The only exception was that each round of sim consisted of a single period in which subjects could make their decision. Eliciting beliefs In both sim and seq Beliefs treatments, after subjects received their private signal, they were asked to state the likelihood that they believed their match would invest in the first period. Subjects were paid for one randomly chosen round.11 In addition, in the seq treatments with e > 0, after they have observed their match’s first period decision, subjects were asked for their best estimate of the state variable, θ , and were incentivized according to a quadratic loss function. Specifically, the payoff was







Q θ e , θ = 100 1 −

1 4e 2





2 θe − θ ,

where θ e was the reported estimate of the state, and θ was the true state.12 Like the other predictions, subjects were paid for one randomly chosen round. 4. Experimental results In this section, we present our main experimental results. All results are pooled across the Beliefs and Regular treatments. We will note any differences that we observe in the analysis of these two treatments whenever necessary. We organize the presentation around a series of results, each of which investigates the validity of the aforementioned hypotheses.

10 11 12

We appreciate a referee’s suggestions, which led to these follow-up sessions. The payoff, in points, was given by 100 − 100(1 − bi )2 1[a j = i] − 100(bi )2 1[a j = w], where a j was the match’s action. In this case, beliefs were restricted to [20, 50] ∩ [xi − e , xi + e ].

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273

Table 3 Basic tests of the theory. seq

sim

Violations of dominance

42 (1021)

6 (818)

Invest in period 2 despite observing w

14 (1432)

na

Invest in period 3

20 (1144)

na

The numbers in parentheses indicate the relevant sample sizes.

Table 4 Individual-level period-one thresholds (last 20 rounds). Treatment

Average threshold

Standard deviation

Average number of mistakes

Fraction of perfect threshold

Equilibrium threshold

p-value (theory)

seq(0) seq(2) seq(5) seq(10)

27.93 28.93 30.61 32.12

3.24 2.13 2.88 7.05

0.17 0.25 0.13 0.81

87.8 87.5 89.3 48.1

27.00a 27.79 28.92 30.74

0.07 0.02 0.00 0.17

sim(0) sim(2) sim(5) sim(10)

27.92 30.49 32.42 33.41

2.50 2.10 3.14 3.67

0.08 0.18 0.38 0.70

91.7 87.5 71.8 46.0

35.00a 35.00 35.00 35.00

0.00 0.00 0.00 0.00

a Observe that there are multiple equilibria in these treatments. In seq(0) any threshold k1 ∈ [25, 27] (with k2 = 27) is an equilibrium and in the sim(0) treatment any threshold κ ∈ [25, 45] is an equilibrium.

4.1. Basic tests of behavior Result 1. The data strongly support Hypothesis 1: (a) Subjects do not play dominated strategies, (b) they invest in period two only if they observe their match invest in period one, and (c) investment takes place in period one or period two. Table 3, which reports the number of failures in three fundamental tests of subjects’ understanding of the problem, provide support for this result. We first look at dominance, where, recall that if a subject’s signal is below 25 − e or above 45 + e, then w and i are dominant actions, respectively. We find that subjects violated dominance only 4.1% of the time in the seq treatment, and only 0.7% of the time in the sim treatment. We next ask whether subjects’ decisions are compatible with the information they receive. Precisely, do subjects invest in period two after observing that their match did not invest in period one? We observe this type of behavior in only 0.98% of all the cases in which a subject who did not invest in period one observes that his match did not invest in period one either. Finally, we see that in only 1.72% of the observations did a subject invest in period three. Of the small number of errors, approximately half took place in the first 10 rounds and about 80% in the first 20 rounds. This indicates that not only these errors are rare, but they also vanish as subjects gain experience. Result 2. In support of Hypothesis 2, a large majority of subjects use monotone strategies. For the seq treatments, the thresholds are higher than the equilibrium prediction, while in the sim treatments, they are lower than the equilibrium prediction. In order to obtain this result we compute the best-fitting threshold for each subject. Specifically, for the first period of the seq treatment, for an arbitrary threshold k1 we say that a subject makes a mistake if he receives a signal, x ≥ k1 but does not invest, or he receives a signal, x < k1 but does invest. We then look for the threshold kˆ 1 that minimizes the number of mistakes.13 If, at kˆ 1 , a subject does not make any mistakes, then we say that the subject has a perfect threshold; that is, he uses a monotone strategy. We repeat a similar exercise for the sim treatment. Table 4 displays the results and the summary statistics of this exercise, where we focus on the last 20 rounds.14 As the fifth column of Table 4 shows, in five of eight treatments, over 85% of the subjects has a perfect threshold. Only in sim(10) and seq(10), did about half of the subjects have a perfect threshold; though even in these cases, the average

13 In the event of a tie, we take the average over all thresholds that minimize the number of mistakes. All of our results hold if we take an alternative tie-breaking rule such as the lowest mistake-minimizing threshold. 14 As noted above, the number of fundamental errors declined substantially after the first 20 periods. If we focused on the entire 40 periods, average thresholds are very similar, but the fraction of subjects using perfect thresholds is significantly lower.

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number of mistakes was less than 1. Therefore, even when subjects did not have a perfect threshold, they were still highly consistent. Table 4 also depicts the theory’s point predictions about thresholds. In the seq game treatments, we see that the average threshold is always above the predicted equilibrium threshold. For seq(0) and seq(10), the difference is not significant at the 5% level, while for the other two seq treatments the difference is statistically significant. In contrast, in the sim treatments, subjects’ thresholds are always significantly lower than the theoretical prediction. Hence, when we take theory as the benchmark, subjects appear not to invest often enough in the seq game but they invest more often in the sim game. Table 4 shows a substantial amount of heterogeneity (as measured by the standard deviation) in thresholds.15 We also see that, heterogeneity increases as the level of noise increases. Also, heterogeneity in seq is more pronounced than in sim treatments. Specifically, including both Beliefs and Regular treatments, the variance of thresholds is higher in the seq game than in the sim game, except when e = 5; however, the difference is only significant for e = 10 (p < 0.01). If we focus only on Regular treatments, then the difference is also significant when e = 0 (p < 0.01). 4.2. seq vs. sim treatment In this subsection we start comparing subjects’ behavior and outcomes in seq and sim treatments in order to test Hypotheses 3 and 4. Result 3. Hypothesis 3 is supported by the data: (a) For e > 0, subjects have significantly lower thresholds, (b) the frequency of coordination on joint investment is significantly higher, and (c) the frequency of miscoordination is lower in the seq treatment than in the sim treatment. In all cases, however, the difference is smaller than the theoretical prediction. Moreover, coordination rates go down and miscoordination rates go up as noise increases. Table 4 qualitatively confirms that average thresholds are generally lower in the seq treatment, except for e = 0, where the inference is clouded by the presence of multiple equilibria.16 To provide statistical support, Table 5 reports the results of an analysis of variance with factors for treatment (i.e., seq vs. sim), noise (i.e., e), and whether or not beliefs were elicited. We find that thresholds are 1.44 points lower in seq, and this is statistically significant ( p = 0.012).17 Therefore, although the difference between the treatments is smaller than the theory predicts, allowing subjects to choose the timing of their investment lowers the threshold for investment.18 When we look at how thresholds vary with the informativeness of signals, we observe that thresholds increase with the noise in both treatments (see Table 4). This observation is consistent with our theoretical prediction for the seq game, yet it is inconsistent with that of the sim game. Indeed, under the assumption of risk neutrality, theoretical thresholds for the sim game do not change with the noise. However, this observation would be consistent with the equilibrium prediction. If we were to assume that the players are risk averse. Although this seems to be a reasonable explanation, in Section 5, we discuss this issue further: Our analysis of beliefs casts doubt on the ability of risk aversion to explain this behavior in the sim treatments. For the seq treatments, we say that a pair of subjects coordinate if they have both invested by the last period, and they miscoordinate if one of them invests while the other never invests. Fig. 2 provides qualitative support for Hypothesis 3(b, c). In the left panel, we see the frequency of coordination in each treatment, while the right panel displays frequencies of miscoordination. In order to provide some statistical support, the second and third columns of Table 5 report the results of an analysis of variance with factors for treatment, noise, and whether or not beliefs were elicited. We find that coordination rates are, on average, 0.113 higher in the seq game, and miscoordination rates are, on average, 0.095 lower in the seq game. In both cases, taking the subject average as the unit of observation, the result is significant at p < 0.01.19 Although it is not formalized as a hypothesis, Table 1(b) indicates that coordination rates should go down, and miscoordination rates should go up as the noise increases.20 The ANOVA results in Table 5 also provide support for this prediction.

15

For example, although not visible in the table, in the seq(10) treatment, estimated first period thresholds range between 24.6 and 44.44. Recall that for e = 0, any threshold k1 ∈ [25, 27] (k1 ∈ [25, 45]) is an equilibrium for seq (sim). 17 One might worry that our ANOVA is mis-specified because it does not include interactions between treatment and noise terms. However, we do not find any evidence for this as we cannot reject the joint hypothesis that the interactions are 0. Therefore, the analysis reported in the main text remains valid. 18 Interestingly, the same ANOVA also suggests that subjects have lower thresholds—on average 1.54 points ( p = 0.018)—in Beliefs treatments than Regular treatments, with most of this difference is attributable to the seq treatment. Therefore, the act of eliciting beliefs may have caused subjects in the seq treatment to think more carefully about the strategic setting, allowing them to come closer to equilibrium behavior. 19 As with Footnote 17, for both the coordination and miscoordination rates, we also estimated ANOVA models in which we included interactions between the treatment and noise values. We draw the same general conclusion but we note that most of the improvement in coordination and miscoordination rates in the seq treatment (relative to the sim treatment) occurs when e ∈ {5, 10}. In both cases, the magnitude and significance of the coefficients on the noise levels increase 20 Note that when e = 0, if subjects can agree on which of the continuum many equilibria to coordinate on, there should be no miscoordination. However, as Fig. 2 depicts, we still observe miscoordination rates of about 5%. It appears that much of this miscoordination arises because of multiple equilibria. For example, in seq(0), of the 20 instances of miscoordination over the las 20 rounds, in 11 of them the signal (and hence state) was between 25 and 27, which coincides with the set of equilibrium thresholds. The remaining 9 instances had signals outside of this range and can, therefore, be viewed as mistakes. 16

F. Brindisi et al. / Games and Economic Behavior 86 (2014) 264–281

275

Fig. 2. Coordination & miscoordination rates (last 20 rounds). Table 5 ANOVA results for thresholds, coordination and miscoordination (last 20 rounds). Factors Treatment sim seq

Noise 0 2

10

Beliefs No Yes

base −1.440** [0.567]

base 0.113*** [0.024]

base −0.095*** [0.016]

na base

base −0.076* [0.042] −0.154*** [0.032] −0.207*** [0.031]

base 0.016 [0.028] 0.055** [0.022] 0.120*** [0.021]

base 0.004 [0.026]

base 0.005 [0.020]

30.420*** [0.753]

N R2

***

Miscoordination

base −1.538** [0.646]

cons

*

Coordination

2.423*** [0.856] 3.576*** [0.824]

5

**

Threshold

228 0.118

0.695*** [0.030] 280 0.212

0.112*** [0.020] 280 0.219

significant at 10%. significant at 5%. significant at 1%.

Coordination rates are higher in the seq game because subjects who choose to wait in period one have an additional period to revise their decision to invest. Therefore, it is important to understand subjects’ behavior in the second period. Unfortunately, there is not enough data to replicate our individual threshold analysis for the second period. Instead, we take a regression approach, similar to that used by Heinemann et al. (2004). Precisely, we specify a logit model that determines the probability that a subject invests in period two after having observed an investment:

Pr(i | x, i) =

exp(a + bx) 1 + exp(a + bx)

,

(1)

where x is the subject’s signal. Note that for x = −a/b, we have Pr(i | x, i) = 1/2; i.e. the subject is indifferent between investing √ and waiting. Hence −a/b is taken as the estimate of the threshold. The standard deviation of the threshold is π /(b 3). Therefore, the parameter b captures the sensitivity to one’s signal. That is, higher values of b indicate sharper thresholds. Table 6 shows that, in all four seq treatments, the estimated average threshold is higher than the theoretical prediction, but the difference is only statistically significant in the seq(5) treatment. There are two possible explanations. First, subjects’ thresholds in period one were generally higher than the theoretical prediction, which means that the signal realizations are necessarily skewed rightward. Second, it appears that although subjects appreciated the value of information contained in observing investment, they may not have updated their beliefs about the state strongly enough.21 When we analyze beliefs in the next section, we will further investigate this latter claim.

21 We say partial because the estimated threshold in seq(10) is less than 27, which is the cutoff for subjects to earn more than by choosing w, and also covering the cost of delayed investment.

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Table 6 Estimated period-two thresholds (last 20 rounds). Treatment

a

b

Mean threshold

seq(0)

−42.49*** (−3.62)

1.56*** (3.61)

27.24

seq(2)

−6.65 (−1.38)

0.22 (1.22)

seq(5)

−9.09** (−2.43)

seq(10)

−3.35*** (−3.59)

S.D. of threshold

Equilibrium threshold

p-value (theory)

1.16

27.00

0.58

30.79

8.40

25.61

0.15

0.28** (2.00)

33.00

6.58

23.54

0.01

0.14*** (4.36)

23.48

12.73

20.13

0.11

Robust z-statistics are in parentheses (clustering at the subject level). * significant at 10%. ** significant at 5%. *** significant at 1%.

Table 7 Efficiency ratios (last 20 rounds). (a) Summary Statistics Noise

0 2 5 10

π a /π max

Comparison

sim

seq

% difference

0.978 0.943 0.913 0.858

0.977 0.963 0.953 0.928

−0.102 2.121* 4.381*** 8.159***

(b) Regression Parameter

Coefficient

|t − stat|

noise (e) e × Dseq

Dseq constant

−0.0118*** −0.0070*** 0.0018 0.9742***

13.76 5.92 0.24 173.66

R 2 = 0.492

N = 308

Note that to facilitate comparison with Table 1(d), we report the percentage difference in the payoff ratio between seq and sim; however, the formal hypothesis tested was: H 0 : seq − sim = 0. * significant at 10%. *** significant at 1%.

Finally, we turn to Hypothesis 4, which states that subjects earn higher payoffs in the seq treatment than the sim treatment, and that the difference between the two treatments is increasing in the noisiness of signals. We have the following confirmatory result: Result 4. Payoffs are higher in the seq treatments, and the difference in welfare between seq and sim is increasing in the noisiness of signals. The hypothesis was stated in terms of the ratio of actual payoff, π a , to the maximum possible payoff a subject could receive, π max = max{25, θ}.22 For each of our treatments, Table 7(a) reports the ratios π a /π max . We see that, except for e = 0, subjects in the seq game earn more than subjects in the sim game, and the difference is statistically significant for all e ≥ 2.23 Notice also that as noise increases, the difference in efficiency between the seq and the sim increases, though it is always less than the theory predicts. Table 7(b) reports the results of the following OLS regression:

π a /π max = β0 + β1 e + β2 (e × Dseq ) + β3 Dseq +  , 22 This provides a more powerful test than simply comparing average payoffs because it is partially controls for potential differences in the distribution of realized values of θ in the two treatments. 0 e 2 23 Letting tdf denote the t statistic when noise is e and the degrees of freedom are df , we have that t 76 = −0.15 (p = 0.878), t 38 = 1.85 (p = 0.073), 5 10 t 86 = 4.01 (p = 0.0001) and t 100 = 7.67 (p = 0.000).

F. Brindisi et al. / Games and Economic Behavior 86 (2014) 264–281

277

Table 8 Average period one beliefs over ranges of signals. (a) seq Treatment Signal range

e=0 seq

[10, 15] (15, 20] (20, 25] (25, 30] (30, 35] (35, 40] (40, 45] (45, 50] (50, 55] (55, 60]

0.087 0.676 0.855 0.931 0.943 0.971

e=5 Theory

0.000 0.600 1.000 1.000 1.000 1.000

seq 0.277 0.188 0.395 0.807 0.886 0.929 0.913 0.869

e = 10 Theory

seq

Theory

0.000 0.000 0.264 0.947 1.000 1.000 1.000 1.000

0.293 0.299 0.405 0.512 0.669 0.783 0.837 0.898 0.915 0.955

0.035 0.152 0.275 0.400 0.580 0.777 0.901 0.976 1.000 1.000

(b) sim Treatment Signal range

e=0 sim

[10, 15] (15, 20] (20, 25] (25, 30] (30, 35] (35, 40] (40, 45] (45, 50] (50, 55] (55, 60]

0.095 0.626 0.852 0.935 0.969 0.991

e=5 Theory

0.000 0.000 0.000 1.000 1.000 1.000

sim 0.079 0.174 0.355 0.613 0.756 0.851 0.859 0.948

e = 10 Theory

sim

Theory

0.000 0.000 0.041 0.289 0.706 0.958 1.000 1.000

0.167 0.183 0.184 0.350 0.532 0.721 0.851 0.916 0.955 0.985

0.000 0.024 0.112 0.222 0.384 0.613 0.776 0.886 0.975 1.000

The highlighted cells indicate those instances in which the average belief is higher than the correct Bayesian belief about the probability their match will invest.

where Dseq is the dummy variable taking value 1 if it is a seq session, and 0 otherwise. We find a negative coefficient on noise, e, and a positive coefficient on the interaction term e × Dseq , both of which are significant at the 1% level. Thus, welfare declines as noise increases but, consistent with the theory, it declines more slowly in the seq treatment. 5. Using beliefs to understand observed behavior While the results presented in the previous section provide support for our hypotheses, we are left with a number of new questions. (1) Why are thresholds higher than predicted by the theory in the seq game but lower than predicted by the theory in the sim game? (2) What is the source of the substantial heterogeneity in thresholds? (3) Why do thresholds vary so strongly with noise in the sim treatment, contrary to the theoretical prediction? (4) Why are average payoffs below the equilibrium prediction in the seq treatment? The Beliefs treatments were conducted in an attempt to answer these questions. 5.1. Beliefs in period one Before we tackle these questions, let us first establish that subjects’ beliefs are consistent with their decisions. In order to do this, we first show that beliefs are monotone in signals, and that subjects’ behavior is consistent with their beliefs—i.e., the probability of investment is increasing in one’s belief that their match will invest. Table 8 reports average beliefs about the likelihood that a player’s match would invest in period one. We pool beliefs across realizations of signals in increments of 5. Table 8 demonstrates that the beliefs are generally monotone increasing in the signal realization, and they also appear to increase more rapidly for lower values of noise.24 Table 9 reports the results of a series of logit regressions where the dependent variable takes value 1 if the subject invested in the first period and the explanatory variables are the signal received and the subject’s stated belief. The table reports marginal effects. Both a subject’s signal about the state and his beliefs about the likelihood of his match investing 24 Comparing the seq and sim treatments, one can see that the beliefs are virtually identical when e = 0 and that for e ∈ {5, 10}, beliefs are generally higher in the seq treatment. Indeed, over some ranges of signals, the difference is statistically significant. In other words, for the same signal, subjects believed that it was more likely their match would invest in the sequential game. This is consistent with the theoretical results and our earlier experimental results.

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Table 9 How beliefs & signals jointly determine investment (last 20 rounds, marginal effects). Parameter

seq(0)

seq(5)

seq(10)

sim(0)

sim(5)

sim(10)

signal

0.0125*** (2.69)

0.0187*** (7.52)

0.0172*** (5.29)

0.0075*** (3.42)

0.0232*** (3.42)

0.0186*** (5.59)

belief

0.0015*** (7.90)

0.0014*** (4.73)

0.0029*** (3.41)

0.0012*** (8.35)

0.0029** (2.17)

0.0034*** (5.71)

Robust z-statistics are in parentheses (clustering at the subject level). * significant at 10%. ** significant at 5%. *** significant at 1%.

Table 10 (Approximatea ) best-response rates (last 20 rounds). Criteria

sim(0)

sim(5)

sim(10)

Best-Response Rate

0.977 (0.041)

0.881 (0.120)

0.900 (0.114)

Frequency of i Conditional on Not BR

1.000 (0.000)

0.804 (0.394)

0.731 (0.446)

Standard deviations in parentheses. a We classified a decision as a best-response if, according to the elicited period one belief, EV i ≥ EV w and the subject invested, or EV i < EV w and the subject did not invest. For the sim treatments, this is the definition of a best-response, while for the seq treatments, as noted in the main text, this measure is only approximate.

have a positive effect on his investment decision. An increase in the signal by 1 point increases the probability of investing by between 0.75 and 2.3 percentage points. The effect of beliefs is much smaller: A one percentage point increase in one’s belief increases the probability of investing by between 0.14 and 0.34 percentage points. The above two results suggest that elicited beliefs are meaningful and part of subjects’ decision making process. With this in mind, we can now use beliefs in our search for answers to the outstanding questions. We first seek to explain the observed differences between predicted and actual period one thresholds. As noted, Table 8 reports the average beliefs over ranges of signals. In columns labeled as “Theory” it also reports the average Bayesian beliefs over the same ranges of signals. We observe that, for a range of signals in the neighborhood of the equilibrium threshold, subjects are overconfident about the likelihood of their match investing relative to the Bayesian prediction. However, when we look at the best-responses that these beliefs generate, this overconfidence does not lead to a change in optimal investment behavior in the seq treatments. However, it often tips the scale in favor of investment in the sim treatments.25 Thus, overconfidence provides a possible explanation as to why thresholds are lower than theory in the sim treatments. Next, we focus on the issue of heterogeneity in thresholds. It appears that heterogeneity in thresholds is at least partially explained by heterogeneity in beliefs. In particular, if we regress the estimated threshold of a subject—reported in Table 4—on the subject’s average belief that his match would invest, then the coefficient is negative and significant: For every percentage point increase in the average belief, the threshold decreases by 0.12 points (p < 0.01). That is, confidence matters: More confident subjects invested at lower thresholds. The relationship is present in both the seq and the sim treatments, though it is stronger and has greater explanatory power in the sim treatments. Next, we want to understand why thresholds in the sim treatment vary with the noisiness of signals. As noted in the previous section, one’s natural inclination is to point to risk aversion. However, as we now argue, we do not find this compelling. In order to demonstrate this, we look at subjects’ best-response behavior. Denoting the belief that the match will invest by b, and the signal by x, we can write the expected payoff from investing as EV i = bx + (1 − b)(x − 20) = x − (1 − b)20 in the sim game. The expected payoff from not investing is always EV w = 25. The subject is best-responding if EV i ≥ EV w and he invests, or if EV i < EV w and he does not invest. Table 10 reports the best-response rates for the sim game. We also report the frequency with which subjects chose to invest conditional on not best-responding to their beliefs. Table 10 shows that, for all noise levels, e, the best-response rate is over 88%. Thus, even though subjects in the sim treatments invest more frequently than the theoretical predictions, their actions are largely consistent with their stated beliefs. We also observe that when subjects are not best-responding to their beliefs, between 73% and 100% of the time they chose to invest when not investing was optimal; that is they choose the risky alternative. This suggests that risk aversion cannot explain why thresholds change when the noisiness of signals increase.

25 For example, in sim(5) a subject with a signal of 32.75 and a (over-confident) belief of 0.613 that her match will invest strictly prefers to invest. Therefore, subject’s over-confidence has made investment optimal for a lower threshold than the theoretical prediction.

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279

Table 11 The impact of observing investment on beliefs (observed vs. theoretical). (a) seq(5) Signal range

Observed Match w

Match i

Diff.

Match w

Theory Match i

Diff.

[18.92, 23.92] (23.92, 28.92] (28.92, 33.92] (33.92, 38.92]

22.385 24.399 30.545 33.213

22.939 26.361 31.505 36.421

0.554 1.963 0.960 3.208

23.066 25.493 28.941 32.273

25.608 28.940 32.451 36.615

2.542 3.447 3.510 4.342

(b) seq(10) Signal range

Observed Match w

Match i

Diff.

Match w

Theory Match i

Diff.

[10.74, 15.74] (15.74, 20.74] (20.74, 25.74] (25.74, 30.74] (30.74, 35.74] (35.74, 40.74] (40.74, 45.74] (45.74, 50.74]

21.644 23.130 25.000 27.396 30.895 37.964 40.120 43.375

21.667 23.908 25.722 28.831 33.695 38.218 42.723 44.920

0.022 0.779 0.722 1.436 2.800 0.254 2.603 1.545

21.574 23.775 25.570 26.734 29.082 32.416 35.746 39.080

22.413 25.747 29.079 32.414 35.852 39.647 42.181 44.210

0.840 1.973 3.509 5.680 6.770 7.231 6.435 5.130

Row shading indicates the range of signals over which it was optimal to choose to wait in period one and, for at least some signals in that range, it would be optimal to invest in period two if the subject observed period one investment by their match. That is, the signal x ∈ (k2 , k1 ).

5.2. Beliefs in period two We now use beliefs in period two in order to answer our fourth question regarding average payoffs in the seq treatment. Of course, one reason why subjects’ payoffs are below the theoretical prediction in the seq treatment is due to inadequate level of investment in period one (as a result of higher thresholds). However, there is an additional factor: beliefs about the state do not respond sufficiently upon observing one’s match invest. We first establish that beliefs about the state influence the probability of investing in period two. A logit regression of the period two decision on signal, belief about the state, and match’s period one investment decision shows that, by far, the largest effect is the decision of the match. Specifically, observing that one’s match invests increases the probability of investment in period two by 26.6%. Hence, despite the fact that subjects do not fully appreciate the value of observing investment, it still is a crucial determinant of the second period decision. Additionally, beliefs matter: For every unit increase in beliefs, the probability of investing increases by almost 1% (and the effect is 2.85% if we restrict the sample to only those who observed period one investment by their match). Both effects are significant at the 1% level. Thus beliefs positively influence the probability of investing in the second period. Now, let us show that beliefs about the state increase after observing an investment. Pooling across noise levels e = 5 and e = 10, we find that if a subject did not invest in period one, his average belief about the state when his match did not invest, was 23.77, while it was 26.91 when his match did invest.26 Moreover, the difference is statistically significant (paired t-test: t 38 = 5.14, p < 0.01). The difference in beliefs about the state is even larger if a subject invested in period one: 32.12 when his match did not invest vs. 39.70 when his match did invest (t 35 = 11.01, p < 0.01). Thus, observing an investment leads to significantly greater optimism about the state. Finally, we show that the increase in beliefs upon observing investment is less than the theoretical prediction. Specifically, for several ranges of signal realizations, Table 11 reports players’ average belief about the state conditional on their matches’ action. As well as the difference in average beliefs when investment is and is not observed. The most relevant range of signals is xi ∈ (k2 , k1 ), where immediate investment is not optimal, but delayed investment is, if they observe their match invest. If beliefs are not sufficiently high in this range, then the subject will fail to invest, despite it being optimal to do so. These rows are shaded in the table. As can be seen, over this range, subjects are overly pessimistic about the state. For example, consider the shaded row in panel (a), which covers the signal range (23.92, 28.92] for the seq(5) treatment. The average belief after observing investment by their match is only 26.398, which makes delayed investment suboptimal since the state would have to be 27 in order to compensate for the cost of delayed investment. However, the theoretically optimal average belief over this range is 28.94—high enough to justify delayed investment. A similar under-appreciation of the signal value of observing investment can be seen in the shaded rows of panel (b), which considers the seq(10) treatment. In conclusion, beliefs are an important determinant of second-period investment behavior, and beliefs do increase upon observing investment. However, the increase is less than the theoretical prediction (i.e., subjects do not fully appreciate the

26 Breaking up the average belief about the state after observing investment by the noise parameter, we see that it is only 26.3 when e = 5, while it is 27.5 when e = 10. Therefore, on average, beliefs about the state are lower than the safe payoff plus the cost of delayed investment when e = 5. This may partly explain the non-monotonicity in period two thresholds (as a function of e) that we reported in Table 6.

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information contained in observing investment). Combined with the inadequate investment in period one, this explains why the effects of endogenous timing of investment are not as strong as the theoretical prediction. 6. Concluding remarks In this paper, we reported the results of an experiment, which tests a global coordination game with endogenous timing in order to understand the effect of timing on coordination under asymmetric information. The equilibrium analysis of the game highlights two forces behind timing decisions: learning and complementarities. In the equilibrium, optimistic players invest first and convey information about the profitability of investment. In addition to this learning effect, the investment eliminates the strategic uncertainty, and encourages investment through strategic complementarities. The normative analysis indicates that endogenous timing leads to full efficiency as players’ private information become infinitely informative. Moreover, the average difference of welfare between the simultaneous and the sequential game increases as signals become less informative. Our experiment confirms many of the main theoretical predictions. Most importantly, endogenous timing of investment is welfare improving, and its benefit, relative to the simultaneous game, is increasing as signals become less informative. Despite this strong concurrence with the theoretical predictions, there is substantial subject-specific heterogeneity and there are instances where behavior and theory do not match. By eliciting beliefs, both about the likelihood that their match would invest in period one, and about the value of the state variable (after observing period one decisions), we are able to gain a deeper understanding of the observed behavior. In particular, a great deal of the heterogeneity in thresholds arises due to heterogeneity in beliefs: Those subjects who are confident that their match will invest, will themselves invest at lower thresholds. Moreover, while thresholds both in the seq and the sim treatments deviate from theory, the vast majority of the time, subjects’ investment decisions are actually best responses to their elicited beliefs. Thus, the significantly lower thresholds in the sim treatment appear to be due to overconfidence about the likelihood that their match will invest (relative to the theoretical prediction). Interestingly, this overconfidence appears to decline as signals became noisier, leading to thresholds closer to the equilibrium prediction. Furthermore, although observing an investment is the decisive factor determining subjects’ period two investment, our analysis of beliefs shows that subjects fail to sufficiently update their beliefs about the state. 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