1 Introduction
Games in which payoffs are complementary in the actions of several agents often admit multiple Nash equilibria if those actions must be chosen simultaneously. When this is true, each equilibrium is supported by a different set of (correct) beliefs about how other agents will play. Because the Nash concept is silent about which set of beliefs, if any, will arise, considerable effort has been devoted to equilibrium refinements that select unique beliefs in a plausible way. For games with common interests, this has led to a fork in the road: refinements that emphasize uncertainty in agents’ beliefs tend to pick the risk dominant equilibrium, while refinements that expand agents’ action sets (for example, by allowing pre-play communication) tend to focus beliefs on the Pareto optimal equilibrium. One naturally wonders who will win the arms race when both belief uncertainty and agents’ attempts to surmount it are factors. This paper explores this question by relaxing the assumption that agents must act simultaneously in a game that is perturbed away from complete information in the manner of Carlsson and Van Damme (1993). It turns out that the option of delaying one’s play, if it is not too costly, helps to prune away inefficient equilibria, even in the presence of incomplete information. When delay is costless, only Pareto optimal equilibria survive. However, the perturbation to information is not innocuous – if delay is costly, then inefficient equilibria are played with positive probability even in the complete information limit.
A simple stylized example may help to convey the intuition behind the results. Suppose Anne and Bob have a tentative plan to go to the theater on Saturday night. After observing different noisy signals about the play’s quality (they read the reviews in different papers) on Friday morning, each must decide independently whether to purchase a ticket or commit to an alternative plan for the evening that does not involve the other one. Suppose that both Anne and Bob prefer seeing the play together rather than alone. Furthermore, suppose that if the quality of the play is good enough (a “triumph”), either agent would prefer to see it regardless of whether the other one goes, and that if the quality is bad enough (a “fiasco”), both would prefer their outside options regardless of what the other one does. The interesting question is what they will do when their signals about the quality of the play are between these extremes, when each would like to go to the play, but only if the other does as well.
The global games literature suggests that this situation can be analyzed via iterated dominance. Suppose Anne’s review says that the play is mediocre. She reasons that there is a reasonable chance that Bob has gotten a fiasco review and will take his outside option for sure. Since she would buy a ticket only if she were almost certain that Bob would do the same, she must take her outside option. But then if Bob’s review says that the play is average, he must worry that Anne may have gotten a mediocre review – if the chance of this is high enough, he should take his outside option too. This process also works from the other end – when Anne reads that the play is merely good, she should nonetheless buy a ticket for sure if the chance that Bob has seen a triumphant review is high enough. Applying iterated dominance from both ends may yield a unique threshold signal above which Anne and Bob buy tickets. Although it would be efficient for them to go to the play whenever it is not a fiasco, under the equilibrium threshold they will go to the play only when the quality is substantially higher. However, as long as the noise in their signals is fairly low, Anne and Bob manage to avoid miscoordination most of the time – only for signals close to the threshold do they face a substantial amount of uncertainty about how the other will act.
Now suppose that there is an additional option to purchase tickets at the door on Saturday, at a slightly higher price. To keep things symmetric, suppose that Anne and Bob can delay committing to their outside options until Saturday for the same incremental cost. Furthermore, Anne and Bob will have a chance to talk on Friday night. If the cost of delay is not too high, then when Anne sees a review near her threshold, she may decide to delay her decision until Saturday in order to see what Bob has chosen. If Bob could be sure that she would delay, then he could lower his threshold substantially, buying tickets for all but the fiascos on Friday, with confidence that she would follow his lead on Saturday. In fact, he will not be quite this sure, but even the possibility is enough to encourage him to reduce his threshold a little bit. This in turn will encourage Anne to reduce her threshold a bit as well. After playing out this cycle of threshold reductions, Anne and Bob will be able to coordinate on efficient play-going considerably more often than they could without the option of delay. In fact, we will show that as noise vanishes and as the cost of delay becomes negligible, they always coordinate efficiently.
Other models incorporating delay and dynamics into the global game setting generally have found weaker or mixed evidence for efficient coordination – see Section 5 for a deeper discussion of the literature. The strength of our results hinges on the small number of agents and on the assumption that committing to an action is observable by other agents. Observability means it is possible for one agent to resolve some strategic uncertainty for others by leading with the efficient action. Small numbers has several implications. First, a leader knows his action will have a large influence on anyone who delays; this reduces his own strategic uncertainty vis-à-vis leading. Second, the option to delay is in part an option to free-ride in the hopes that someone else will take a decisive action that can be followed. Thus, an endogenous choice to delay in effect creates a secondary coordination problem to do with who, if anyone, will act first and set an example – if the incentive to free-ride is strong enough, delay might not help much with the primary coordination problem. When the number of agents is small, the incentive to free-ride is also small, and so it is not only possible but an equilibrium outcome for agents to choose to lead with the efficient action.
The assumption of observable actions can be particularly natural in many settings where the number of agents is small. For example, consider a small number of nations attempting to coordinate action to address climate change. If one of these nations commits by enacting legislation before the others, this will generally be a matter of public record. Or consider a potential joint venture that requires sharing sensitive information with the prospective partner. Releasing that information to the potential partner unilaterally is, by definition, an observable commitment toward efficient coordination. Alternatively, consider an incident where a few bystanders witness a misdeed (a mugging, or an act of bullying, perhaps) that they could stop if enough of them act. Then if one of them steps forward first to intervene, this will clearly be observable by the others.
The next section describes the formal models first without, and then with, the option of delay, and in the latter case derives conditions under which all equilibria exhibit delay. In such an equilibrium, there is a positive probability that both agents delay their decisions. When this happens, it becomes common knowledge that the dominance arguments above do not apply, and so the potential for multiple equilibria re-emerges. Section 3 characterizes two prominent equilibria: an “optimistic”’ equilibrium in which the agents expect to coordinate on the efficient action in the continuation after both have delayed, and a “pessimistic” equilibrium in which they expect coordination on the inefficient action. In both cases, the gap between equilibrium play and efficient play vanishes as the cost of delay shrinks, but in the “optimistic” equilibrium, convergence to efficiency is at a faster rate. Section 4 provides a similar convergence result for all equilibria. The value-added in the general result is that it places no a priori restrictions on the form of strategies; in particular, threshold strategies are not assumed. Section 5 concludes with a discussion of these results in the context of other recent work on strategic timing in games with complementarities.
2 The Model
Before developing the full model, we begin by introducing a simpler version without delay called the benchmark game and sketch how iterated dominance can be applied to identify a unique equilibrium of this game. Then we add the delay option to the model and establish a preliminary result showing that this option will be used in equilibrium.
2.1 The Benchmark Model
The benchmark game (BG) that we will look at is a special case of a global game as defined by Carlsson and Van Damme. It resembles the game that has been used by Morris and Shin (1998) among others to study currency attacks. There are two agents who simultaneously decide whether to invest (I) or not (N). An agent who invests incurs a cost of a and earns a return of vH if the other agent also invests and vL if he does not, with vH>vL>0. An agent who does not invest always earns 0. Under complete information, this situation is summarized by the following game form.
| I | N |
| I | vH − a, vH − a | vL − a, 0 |
| N | 0, vL − a | 0, 0 |
For moderate investment costs (a∈(vL,vH)), this game has two strict pure strategy Nash equilibia: the Pareto optimal one in which both agents invest, and another one in which neither does.
Next we introduce incomplete information about the investment cost. Assume that it is common knowledge that the investment cost is a random variable a˜, with realization a, distributed uniformly on [vL−ϵ,vH+ϵ]. Each agent receives a private noisy signal of the cost ai=a+ϵhi, where h1 and h2 are independent draws from a symmetric, strictly increasing, atomless distribution H with mean zero and support on [−1,1]. Write A=[vL−2ϵ,vH+2ϵ] for the space of possible signals, and observe that a strategy for agent i now must specify an action for each signal he might receive.
We call this environment the benchmark game.
We will show that BG has a unique symmetric Nash equilibrium that can be identified by iterated dominance. The argument goes as follows. Define thresholds a_i and aˉi to be the highest signal below which agent i always chooses I and the smallest signal above which i always chooses N respectively. These thresholds must exist because for small (large) enough ai, investing (not investing) is a dominant strategy. Now consider agent 1’s expected payoff to investing when he receives the signal a1=aˉ2. Because of the symmetry of the error distributions, he believes it equally likely that agent 2’s signal is higher or lower than his. Since agent 2 never invests for a2>aˉ2, agent 1 expects agent 2 to invest with probability no greater than 12. Thus his expected profit upon investing is
π1(I | a1=a¯2)=Pr(2invests)vH+(1−Pr(2 invests))vL−a¯2≤(vH+vL)/2−a¯2
Furthermore, agent 1’s expected profit from investing with signals higher than aˉ2 is certainly less than this, as the expected cost will be higher and the probability of co-investment lower. If (vH+vL)/2−aˉ2<0, this means that agent 1s expected profit from investing is strictly negative for all a1≥aˉ2. We can conclude then that if aˉ2>vˉ=(vH+vL)/2, then aˉ1<aˉ2. However, by switching the agents in this argument, it is also true that if aˉ1>vˉ, then aˉ2<aˉ1. This leads to a contradiction if aˉ1 and aˉ2 are both greater than vˉ, so in a symmetric equilibrium, we must have aˉ1=aˉ2≤vˉ.
Next consider agent 1’s expected payoff to investing when his signal is a1=a_2. In this case, he expects co-investment with probability at least 12 since agent 2 always invests when a2<a_2. Thus his expected payoff to investing is at least vˉ−a_2. Since this is strictly positive when vˉ>a_2, by reasoning parallel to that above, we can conclude that vˉ>a_2 implies a_1>a_2. Reversing the agents again, we also have vˉ>a_1 implies a_2>a_1. To avoid a contradiction, we must have a_1=a_2>vˉ in a symmetric equilibrium. But then because the lower threshold cannot be greater than the upper threshold, we must have a_i=aˉi=vˉ. Thus the (iterated) dominance regions for N and I completely partition the set of signals, and the equilibrium is unique: invest if and only if the cost signal ai is less than vˉ.
Several points are worth noting. First, as ϵ vanishes, the agents coordinate with probability 1. Moreover, they coordinate on the equilibrium that is risk-dominant given the cost realization, so the effect of strategic uncertainty persists even as that uncertainty vanishes. Moreover, this effect depends crucially on the fact that no matter how small ϵ is, it is never common knowledge that both equilibria are possible (i.e., that neither strategy is dominant).
2.2 The Model with Delay
In this section, the benchmark model is augmented by assuming that agents need not always act simultaneously. Now an agent may either pick an action immediately or defer the decision until later. The timing is as follows. First, agents observe their private signals. Then, in period 0, the agents simultaneously decide whether to invest, not invest, or wait (W). If neither agent waits, the game ends immediately, with payoffs as before. Otherwise, each agent choosing to wait incurs a cost of c and the game moves to period 1. In period 1, the agents who have waited observe the actions taken in period 0 and choose either I or N (simultaneously, if both have waited). As soon as both players have chosen an action from {I,N} payoffs are realized as before. Call this the asynchronous game (AG).
Two points are worth emphasizing. First, the actions N and W are not equivalent because an agent choosing N does not preserve the option of choosing I later. For this reason, it may be best to think of N as a commitment to invest in an alternative project with constant payoff 0. When we refer to “not investing” in the sequel, this interpretation should be borne in mind. Second, payoffs are realized once and for all (not period by period) and depend only on the final actions taken by the agents and any relevant waiting costs. To fix ideas, one may think of two firms positioning themselves to enter one of two markets (I or N) that will open in period 1. Committing to one of the markets in period 0 allows a firm to spread out the investment costs associated with entry, saving c, but a firm can also delay its choice until period 1. Profit flows are realized once and for all when the markets open at period 1 and depend only on whether both firms are in market I (but not on when they decided to enter).
It is clear that if the waiting cost is large enough that delay is a dominated strategy, then the equilibria of AG and BG coincide. The next result establishes a rough converse: delay always occurs when it is relatively costless. For brevity, define Δv=vH−vL.
Proposition 1Ifc<Δv/4, then every sequential equilibrium of AG has a positive probability of delay.
ProofSuppose to the contrary that there is an equilibrium in which each player chooses I or N in period 0 for every signal. Then the same argument used to establish the BG equilibrium implies that each agent invests precisely when his signal is greater than vˉ. Thus, the payoff to agent 1 when his signal is a1=vˉ is 0. Were he to wait, he would observe 2 choosing either I (if a2<vˉ) or N (if a2>vˉ), events he believes to be equally likely. By waiting and mimicking 2, he earns 12⋅0+12(vH−E(a˜|a2<a1=vˉ))−c>(vH−vˉ)/2−c=Δv/4−c, a profitable deviation.■
This proposition underscores the fact that uncertainty about one’s opponent’s action has two effects: it encourages the choice of “safe”actions, but it also creates an option value to delaying one’ decision until the uncertainty is resolved. The BG equilibrium compresses all strategic uncertainty into a small region around the threshold signal, creating strong incentives for agents with signals in that region to wait. On the other hand, agents with extreme signals are sufficiently sure of how their opponents will act that waiting is not worthwhile. In the sequel, we assume that c<Δv/4 and show that equilibria exist in strategies with two thresholds for period 0 action: a signal below which the investment is immediately made and a signal above which the decision not to invest is made immediately, with waiting in the intermediate region.
3 Simple Equilibria
One implication of Proposition 1 is that in a symmetric equilibrium, there is a positive chance that both agents wait in period 0. We will refer to the pair of strategies in period 1 after both agents have waited in period 0 as the continuation game. A key observation is that it must be common knowledge in the continuation game that neither agent’s signal is too extreme. For example, if the equilibrium specifies that each agent always plays I (N) immediately for signals less than a_ (greater than aˉ), then upon arriving in the continuation game it is common knowledge that both signals lie in [a_,aˉ]. If this is a subset of [vL,vH], then it is common knowledge that neither agent believes either strategy to be ruled out by dominance. As a result, the continuation game will typically have multiple equilibria. This casts some doubt on whether it will be possible to make sharp predictions about outcomes of AG.
The equilibria constructed below go some way toward answering this question. Both are characterized by simple threshold strategies (a_,aˉ) under which an agent invests immediately for cost signals below a_, chooses N immediately for cost signals above aˉ, and waits for intermediate signals. The first equilibrium assumes an “optimistic”continuation game in which the agents coordinate on investment after both wait, while the second assumes a “pessimistic”’ continuation in which there is no investment. The equilibria share certain qualitative features. In both cases, the waiting region shrinks as signals grow more precise, with immediate coordination in the noiseless limit. Furthermore, in both cases this coordination is more and more frequently on the efficient investment I as the cost of waiting becomes small.
3.1 An “Optimistic” Equilibrium
We will look for a symmetric equilibrium in threshold strategies as described above. In addition to the thresholds (a_i,aˉi), a full strategy description must specify the action taken in period 1 by an agent with signal ai who has observed his opponent choose S∈{I,N,W} in period 0.
As a first step, observe that an agent who has waited and observed his opponent choose I or N in period 0 will always follow suit. To see why, suppose that in some equilibrium, agent 1 were to choose N after observing 2’s choice of I in period 0. This can only happen if 1’s posterior belief about a˜, revised to incorporate the fact that a2≤a_2, makes N a dominant strategy for him. However, 2’s choice of I is the most favorable news about I that 1 could possibly receive, so he must expect to play N regardless of what he observes in period 0 after waiting. In this case, he is better off playing N immediately and saving the waiting cost. The same is true for the opposite case.
Now consider the situation in which both agents have waited until period 1. At this point, it is common knowledge that a1∈W‾1=[a_1,aˉ1] and a2∈W‾2=[a_2,aˉ2]. Furthermore, it is common knowledge that 1’s expectation of a˜ is less than E(a˜|a1=aˉ1, a2∈W‾2), so whenever aˉ1≤vH and aˉ2≤vH, it is common knowledge that 1 does not believe N to be a strictly dominant strategy. Extending this argument, when W‾i∈[vL,vH] for i∈{1,2}, it is common knowledge that neither player believes either strategy to be strictly dominant.
For the time being, we will assume that this condition holds. Consequently, dominance arguments do not restrict the set of equilibrium outcomes in this continuation game. Here we focus on the optimistic continuation in which both agents always choose to invest in period 1 after.. is played in period 0.
Next we turn to identifying optimal period 0 thresholds (a_,aˉ) given this continuation. Toward this end, let us suppose that agent 2 uses the strategy (a_,aˉ) and consider agent 1’s best response. In a symmetric equilibrium, we will need agent 1 to be indifferent between I and W when a1=a_ and to be indifferent between W and N when a1=aˉ. This pair of indifference conditions will pin down the equilibrium thresholds. Before jumping into the analysis, it will be helpful to define the distribution of the difference between the errors in the agents’ signals G (normalized by ϵ); that is, h1 – h2 G. The symmetry of H implies that G is symmetric with mean 0. We let ϕ(k)=1−G(k) be the probability that the difference in the errors exceeds kϵ. Given the agents’ uniform prior over a˜, we can then express the probability that agent 1 places on agent 2 having received a signal substantially higher or lower than his own signal
:
Pr(a2<a1−kϵ|a1)=ϕ(k)Pr(a2>a1+kϵ|a1)=ϕ(k)
In particular, Pr(a2<a1|a1)=Pr(a2>a1|a1)=ϕ(0)=1/2.
Now we turn to agent 1’s best response. Regardless of his signal, he can earn 0 by choosing N. Let us write πI(a1) and πW(a1) for the expected payoff he earns by choosing I or W respectively when his signal is a1. If he chooses I, he will succeed in coordinating with agent 2 whenever agent 2 chooses I or W – that is, whenever a2<aˉ. He will fail if agent 2 chooses N, which happens when a2>aˉ. The probability of this latter event is Pr(a2>a1+(aˉ−a1)|a1)=ϕ(k), where k=(aˉ−a1)/ϵ. Thus, the expected payoff to choosing I is
πI(a1)=(1−ϕ(k))(vH−E(a˜|a1;a2<aˉ))+ϕ(k)(vL−E(a˜|a1;a2>aˉ))=vH−ϕ(k)Δv−E(a˜|a1)
Alternatively, if agent 1 were to choose W, he would still end up coordinating on I whenever agent 2 chooses I or W. (In the first case he would follow agent 2’s action, and in the second case we have assumed an optimistic continuation.) The difference is that he can avoid miscoordinating when agent 2 chooses N. His expected payoff is
πW(a1)=(1−ϕ(k))(vH−E(a˜|a1;a2<aˉ))+ϕ(k)(0)−c=(1−ϕ(k))(vH−E(a˜|a1;a2<aˉ))−c
where the second term in the first line is written explicitly to emphasize the case in which agent 1 avoids miscoordinating by waiting. Now we would like to show that there exist (a_,aˉ) such that agent 1’s best response to (a_,aˉ) is (a_,aˉ). Essentially, this is a matter of showing that Figure 1 accurately represents the relationship between πI(a1), πW(a1), and 0. More formally, we need to show that there exist (a_,aˉ) such that
πI(a1) is strictly decreasing.
πW(a1) is strictly decreasing and crosses 0 at aˉ<vH.
πI(a1)−πW(a1) is strictly decreasing and crosses 0 at a_, with vL<a_<aˉ.
These claims are verified in the appendix for sufficiently small noise. Observing that ϕ0=12, the indifference condition πW(aˉ)=0 yields
[1]12(vH−E(a˜|a1=aˉ,a2<aˉ))=c
while the condition that πI(a_)=πW(a_) gives us
[2]ϕ(kˉ)(E(a˜|a1=a_,a2>aˉ)−vL)=c
where kˉ=(aˉ−a_)/ϵ.
This construction allows us to state the following result.
Proposition 2Whenc<Δv/4andϵ<2c, a (symmetric) sequential equilibrium of AG exists with threshold strategies defined by eqs (1) and (2) and an optimistic continuation.
The condition on ϵ is imposed to ensure that the waiting region (a_,aˉ) lies strictly within (vL,vH). For positive ϵ, there is a chance that an agent whose signal lies just outside (vL,vH) will have her beliefs about a˜ shifted into (vL,vH) by her opponent’s period 0 action. Thus, her option value of waiting is positive; if the waiting cost is sufficiently low, she will not act immediately. With a bit more work one can show that the equilibrium we have constructed does not change much in this case, but we will not present this since the emphasis later will be on limiting equilibria with vanishing noise and finite waiting costs. In order to illustrate the intuition behind conditions (1) and (2), we present an example in which the waiting region (a_,aˉ) can be explicitly characterized.
Suppose that H is a uniform distribution on [−1,1]. Then the distribution of the difference in signals will have a triangular density with support on [−2ϵ,2ϵ]. One can show that E(a˜|a1=aˉ,a2<aˉ)=aˉ−ϵ/3, so using eq. (1), the upper threshold is simply aˉ=vH−2c+ϵ/3. In other words, the range of signals for which an agent immediately chooses not to invest despite believing that joint investment is Pareto optimal, [vH−2c+ϵ/3,vH], is positive and shrinks as the cost of waiting declines. This range also shrinks as uncertainty about the cost of investing grows; this is because greater uncertainty about one’s opponent’s signal results in greater uncertainty about his action, increasing the option value of waiting. Because the outcomes of I and W coincide except when one’s opponent chooses N, the lower threshold depends on the probability of this happening. Here the expressions are more complicated:
Pr(a2>aˉ|a1=a_)=ϕ(kˉ|a_)=18(2−kˉ)2E(a˜|a1=a_,a2>aˉ)=a_+4/3−kˉ2(1−kˉ/3)(2−kˉ)2ϵ
and the equilibrium a_ solves a cubic equation. The important point though, is that an agent with signal a_ must expect his opponent to choose N with strictly positive probability, ϕ(kˉ) – otherwise there would be no option value to waiting and he would do strictly better by investing immediately. Thus, kˉ=(aˉ−a_)/ϵ<2, so the width of the waiting region cannot be greater than 2ϵ.
The Noiseless Limit
Now consider what happens upon approaching the complete information limit, holding the waiting cost fixed. As ϵ→0, eq. (2) goes to
18(2−kˉ)2(vL−a_)=c
so in the limit, kˉ must be less than 2, and aˉ−a_=kˉϵ must go to 0. In other words, because the width of the waiting region is constrained by the noise in the signals for the reasons discussed above, the waiting region must vanish along with that noise. Then, in the limit, a_=aˉ=vH−2c, and the agents always coordinate immediately – on (I,I) when a<vH−2c, and on (N,N) when a>vH−2c. Even though the option to wait is not exercised in the complete information limit, its influence does not disappear: as the waiting costs shrinks from Δv/4 to 0, the threshold between coordination regimes grows from vˉ, its level in the benchmark game, to vH. If we let both the noise and the delay costs vanish (with the former going to zero first), then the agents coordinate on immediate investment whenever it is efficient to do so.
Furthermore, the limiting behavior of this equilibrium does not depend on the distribution of the noise; for general H we have the same result.
Proposition 3In the limit asϵ→0of the symmetric equilibrium described above, (I,I)is played immediately ifa<vH−2cand(N,N)is played immediately otherwise. Asc→0, (I,I)is played whenever it is efficient.
ProofFix a positive c. As in the case where H is uniform, we refer to conditions (1) and (2). As ϵ→0, a1 and a2 converge to a, so E(a˜|a1=aˉ,a2<aˉ)→aˉ. Then eq. (1) implies that aˉ→vH−2c. Next, if aˉ and a_ were to differ by more than 2ϵ, the probability of observing signals more than aˉ−a_ apart would be 0, as would be the left-hand side of eq. (2). Since this is inconsistent with eq. (2), aˉ−a_ must go to 0 with ϵ.■
The form of this particular equilibrium is not entirely robust when limits are taken in the opposite order. To see why, refer to the case with uniform noise. Fixing ϵ and taking c to 0, the equilibrium conditions give us aˉ=vH+ϵ/3, or in other words, there are agents who believe N to be a dominant strategy but nonetheless wait. By itself this does not pose a problem; it is simply a statement that the option value of waiting is positive. However, the equilibrium conditions ensure that E(a˜|a1=aˉ,a2<aˉ)=vH – that is, with a zero waiting cost, an agent who is just indifferent between N and waiting expects the value of joint investment to be zero conditional on it occurring. The event he conditions on is a2<aˉ because he expects joint investment to occur whenever 2 does not choose N immediately. However, upon arriving at period 1, he has additional information about 2’s action in period 0. His revised expectation of the investment cost is either E(a˜|a1=aˉ,a2<a_)<vH after observing I or E(a˜|a1=aˉ,a2∈[a_,aˉ])>vH if 2 waited. Thus he is ex post unwilling to play I after observing the (relatively) bad news that his opponent waited also, and the optimistic continuation cannot be sustained.
This equilibrium could be patched up to be robust to the order of limits, but it would still suffer from the criticism that assuming an optimistic continuation after both agents wait is arbitrary. The next section investigates whether any of the flavor of this equilibrium is preserved when the assumption of an optimistic continuation is dropped.
3.2 A “Pessimistic” Equilibrium
The construction is very similar to the previous one with the following exception: now when both agents have waited in period 0, the continuation strategies will specify that they both choose N in period 1. This will change the indifference conditions that determine the waiting region.
First consider the upper threshold. In the optimistic equilibrium, if agent 1 chooses to wait he can expect the final outcome to be (I,I) if a2<a_ or a2∈(a_,aˉ) and (N,N) if a2>aˉ. With a pessimistic continuation, he expects (I,I) only if a2<a_ and (N,N) otherwise. This makes waiting less attractive and changes the indifference condition between W and N to
[3]ϕ(kˉ)(vH−E(a˜|a1=aˉ,a2<a_))=c
The effect will be to tend to push down aˉ relative to the optimistic case.
Next we look at the lower threshold. If agent 1 invests, he expects the outcome to be (I,I) if a2<aˉ and (I,N) otherwise. Alternatively, if he waits, he expects (I,I) if a2<a_ and (N,N) otherwise. Noting that Pr(a2<aˉ|a1=a_)=1−ϕ(kˉ) and Pr(a2<a_|a1=a_)=12, the condition for indifference between I and W becomes
[4](1−ϕ(kˉ))vH+ϕ(kˉ)vL−E(a˜|a1=a_)=12(vH−E(a˜|a1=a_,a2<a_))−cϕ(kˉ)Δv−12(vH−E(a˜|a1=a_,a2>a_))=c
(The second step applies the law of iterated expectations to E(a˜|a1=a_).) For the pessimistic equilibrium, an additional regularity condition on the distribution of the noise is needed.
Condition 1G is weakly log-concave.
Proposition 4A (symmetric) sequential equilibrium of AG exists with threshold strategies defined by eqs (3) and (4) and a pessimistic continuation.
ProofThe proof, which is in the appendix, involves showing first that thresholds satisfying eqs (3) and (4) exist and are unique, and then that using these thresholds is optimal for each agent.■
As before, the extent of delay is limited by the degree of uncertainty about the investment cost: at each threshold, an agent will only be indifferent between acting immediately and waiting if there is a positive probability that his opponent will take the opposite action to the one he is considering; consequently, the size of the waiting region aˉ−a_ is bounded above by the magnitude of the signal noise. (That is, ϕ(kˉ|a)>0, and so aˉ−a_<2ϵ.) The limit properties of this equilibrium hinge, as earlier, on this fact.
The Noiseless Limit
Because of the bound above, aˉ−a_→0 as ϵ→0, so in the limit as the noise in signals vanishes, the equilibrium strategies can be characterized by a single threshold aˉ. To find this limit threshold, we take the limit of conditions (3) and (4) as ϵ→0, yielding
φ(k¯ | a¯)(vH−a¯)=cφ(k¯ | a¯)Δv−12(vH−a¯)=c
Solving for the threshold produces
[5]vH−aˉ=c2+2cΔv−c
It is worth pausing for a moment to compare the limit equilibria generated by optimistic and pessimistic continuations. In both cases, there is immediate coordination with probability 1 as signal noise vanishes. Furthermore, in both cases, the range of costs (aˉ,vH) for which the agents fail to coordinate on investment even though it would be Pareto optimal to do so shrinks as the cost of delay declines. Finally, both cases are outcome equivalent to the benchmark game when c≥Δv/4 and produce fully efficient coordination as c→0.
However, they differ in the rate at which the rate at which inefficient coordination vanishes with c. For the optimistic limit equilibrium, this region vanished at rate c, but for the pessimistic limit equilibrium, eq. (5) indicates that it shrinks at the slower rate c1/2. In understanding this difference, it may be helpful to think about how these equilibria might arise out of a sequence of deviations from strategies that do not use delay.
Start with the unique equilibrium of BG, in which I (N) is played for signals below (above) vˉ. For finite ϵ, these strategies leave agents with signals near vˉ uncertain about how their opponents will play (as demonstrated in Proposition 1), giving them an incentive to wait. For the sake of clarity, suppose that only agent 1 is allowed to wait. Then agent 1 should deviate to waiting in some neighborhood of vˉ. Now consider agent 2’s best response to this deviation. Now when he chooses I, joint investment will occur more often than before, because there will be some signals a1>vˉ for which 1 previously would have played N, but now will wait and follow 2. This makes I more attractive to agent 2, and she should react by shifting her threshold between I and N upward, toward higher costs. But this in turn makes I more attractive to agent 1, and he should respond by shifting his waiting region upward, thus eliciting another response by agent 2, and so on. The upward march of the two agents’ thresholds will continue as long as the option value to agent 1, minus c, is greater than zero (the payoff to choosing N). This option value is roughly equal to the probability that agent 2 chooses I∼12 times the gain to coordinating on I (roughly vH−aˉ). Setting (vH−aˉ)/2≈c yields aˉ≈vH−2c.
The logic is similar when both agents can wait. With an optimistic continuation, the option value to waiting at aˉ (relative to choosing N) is still roughly (vH−aˉ)/2 because the chance of an opponent choosing either I or W is 12 and both lead to coordination on I. As with one-sided waiting, this should lead to a threshold of roughly aˉ≈vH−2c.
When conjectures are pessimistic, the option value to waiting at aˉ is lower. While the gain to coordinating on I is still roughly (vH−aˉ), this will happen only when the opponent chooses I (rather than I or W), and the probability of this is less than 12. To identify this probability, call it p, more precisely, we can use information about the lower threshold a_. At the lower threshold, choosing I rather than W leads to a gain of about vH−a_≈vH−aˉ if the opponent chooses W and a loss of a_−vL if the opponent chooses N (call this latter chance q). As a_ and aˉ increase, the gain vH−aˉ decreases. Since a_−vL does not decrease, indifference requires that q decreases at roughly the same rate as (vH−aˉ). But finally note that p and q are equal – each is the probability that the opponent and own signals differ by more than aˉ−a_. Going back to where we started, this means that the option value to waiting at aˉ is on the order of p(vH−aˉ)≈(vH−aˉ)2. This leads to a threshold that approaches vH more slowly (aˉ≈vH−c) as the cost of delay declines.
4 General Results
As we have seen, by adding delay to the benchmark game, we lose one of the major selling points of the global game approach – the equilibrium prediction is no longer unique. A natural concern is that the equilibrium set may be too large to make any robust predictions about game outcomes. The examples in the last section provide some hope that there may be some properties shared by all equilibria; this section will provide a general result that partially confirms this hope. Before proceeding to the result, we offer two exemplary equilibria that illustrate why general results may be difficult to obtain.
Example 2“Non-monotonic” Equilibrium
Suppose that errors follow the same uniform distribution as in Example 1, and set the model parameters to vH=1,vL=0,ϵ=.01,c=.02. Define the following partition of the signal space into seven regions Ri=(αi−1,αi), with 0=α0<α1<...<α7=1. The other region boundaries are presented below.
α1=0.81383α2=0.82447α3=0.83687α4=0.85252α5=0.94742α6=0.96333
The following strategy specifying how to act depending on which region one’s signal falls in, constitutes a symmetric equilibrium:
R1R2R3R4R5R6R7IWNNWIIWIN
In this table, WI is shorthand for “Wait in period 0. If the opponent waits as well, choose I in period 1, otherwise follow the opponent’s action.” WN is defined similarly.
Figures 2 and 3 show the expected payoff to agent 1, as a function of his signal, for each action when agent 2 is playing according to the strategy above. Figure 3 provides a closer look at the “non-monotonic” interval R1−R5 in which the best response is to invest for low and high cost signals but choose N for intermediate signals.
One can think of this as a regime-shifting equilibrium in which the agents coordinate on the pessimistic equilibrium of 3.2 for low cost signals and coordinate on the optimistic equilibrium of 3.1 for high cost signals. The two regimes are separated by a buffer region R4 of signals for which the agents wait. By way of comparison, the complete information game (ϵ=0, c>0) admits as a correlated equilibrium any arbitrary partition of the public signal a into sunspot regions, with coordination on I or N depending on the region. Noise in the public signal (ϵ>0) disrupts the pinpoint precision required to sustain arbitrary sunspot regions as an equilibrium, but the buffer provided by the waiting regions allows a certain degree of sunspottiness to survive. To highlight one example of how this buffer operates, consider the border signals α4 and α5. At both signals, an agent must be indifferent between investing immediately and waiting; however, investing is less attractive at α5 because the cost is about 10% higher. Compensating for this is the fact that the agent at α5 faces a smaller risk of being stranded by an opponent playing N than an agent at α4. (The probabilities are 0.0209 and 0.0237 respectively.) This compensating reduction in risk is made possible by the flexibility of the waiting regions: R6 provides a wider buffer from the N−playing regions than R4 does.
Equilibria such as this one may offend our aesthetic taste for simplicity, and they are certainly tedious to compute; categorizing all the possibilities does not appear to be a practical option. Whether we can afford to ignore this sort of equilibrium as a possible description of reality is less clear. For example, financial professionals often claim that market dynamics are characterized by resistance, barriers, and thresholds, terms that sometimes seem to describe non-monotonic behavior and generally only make sense if sunspots are at work. The example is constructed so that by excising the sunspot – that is assigning N to be played on {R4,...,R7} without making any changes to the strategy on {R1,R2,R3} – we arrive back at the pessimistic equilibrium of Section 3.2. One might be led to conjecture from this that no equilibrium does worse than the simple pessimistic equilibrium in generating investment; however, the next example shows that this is incorrect.
Example 3Partial coordination in the continuation game
All of the equilibria constructed thus far have specified full coordination on either I or N in the continuation game in each waiting region. This example illustrates that partial coordination is also a possibility. The setup is as in the previous example except that we will look at a wider range of values for c and ϵ. The proposed strategy has the form
R1R2R3R4IWIWNN
where R1=(0,a_), R2=(a_,a∗), R3=(a∗,aˉ), R4=(aˉ,1). Thus, after both agents have waited, neither can be sure how the other will act. Agents with relatively low cost signals will invest, those with high cost signals will choose N, and sometimes the agents will fail to coordinate. There are now three equilibrium conditions: one at a_, one at aˉ, and an equation determining the threshold a∗ at which the agents switch from WI to WN. The condition at aˉ is identical to the one for the pessimistic equilibrium (3). However, the condition at a_ differs from eq. (4) because the chance of co-investment when the opponent’s signal lies in R2 makes waiting more attractive than it would be if WN were played on both R2 and R3. If we write k′=(a∗−a_)/ϵ, so that Pr(a2>a∗|a1=a_)=ϕ(k′), then the condition at a_ becomes
ϕ(kˉ)E(a˜|a1=a_,a2>aˉ)−ϕ(k′)−ϕ(kˉ)=c
At a∗ we must have indifference between WI and WN. The two strategies differ only in the event that both agents wait. In this case, we can write p(a)=Pr(a2∈R2|a1=a) for the probability that agent 2 invests when a1=a. Note that p(a) is decreasing in a. Furthermore, we can write A(a)=E(a˜|a1=a,a2∈{R1,R2}) for the expected cost incurred by investing with signal a1=a after both agents have waited – this is increasing in a. The threshold a∗ between WI and WN is defined by the unique solution to
(1−p(a))vL+p(a)vH−A(a)=0,orp(a)=A(a)
Expressions for p(a) and A(a) are derived in the appendix. Figure 4 presents the equilibrium thresholds when the waiting cost is equal to the level of noise, computed for several values of c=ϵ between zero and 0.2. The solid lines represent a_ and aˉ; the horizontal distance between them is the waiting region R2 ∪ R3 for a particular level of c and ϵ. (The boundary between R2 and R3 is omitted.) For comparison, the dashed lines indicate the corresponding waiting region for the pessimistic equilibrium of Section 3.2. The two equilibria are quite close when c and ϵ are large, but for small c and ϵ, the range of investment costs for which the partial coordination equilibrium generates investment converges to [0,1] more sluggishly than for the pessimistic equilibrium. The problem begins with the lower threshold a_, where a switch from WN to WI on R2 tends to make waiting more attractive, pushing a_ down. This switch does not have a first order effect at the upper threshold aˉ, as agents here never invest after waiting. The second order effect of a_ shifting down is to make waiting less attractive at aˉ. (The chance of observing the opponent choose I in period 0 declines.) This tends to push aˉ down. The overall effect is to shift the entire waiting region down, toward lower investment costs.
In contrast with the earlier examples, the limiting behavior illustrated in Figure 4 applies when c and ϵ go to zero at the same rate (rather than letting ϵ vanish first). Nonetheless, the earlier result of convergence to investment for all a∈(0,1) is still obtained, suggesting that the prior results are not driven by the order of the limits. For technical reasons, the general result below is developed for the ordered double limit (first ϵ→0, then c→0), but the fundamental logic of the proof appears to survive when c and ϵ vanish at the same rate.
In all of the equilibria we have presented, as noise vanishes, there is a threshold cost below which investment always occurs, and this threshold approaches the efficient level vH as the cost of waiting goes to zero. In the rest of this section, we prove that two broad classes of equilibria of AG share this feature. We start with some notation. Let us denote by AG(ϵ,c) the game with level of noise ϵ and cost of waiting c. We say that a strategy is a single waiting region (SWR) strategy if it specifies some a_ and aˉ such that I is played below a_, N is played above aˉ, and some combination of WI and WN is played on (a_,aˉ). A strategy is simple if every contiguous region on which a version of W is played contains only WI or WN. The equilibrium strategy of Example 2 is simple but not SWR, while the strategy of Example 3 is SWR but not simple. The equilibrium strategies from Section 3 are both simple and SWR. Let Γϵc (Ψϵc) be the set of symmetric, sequential simple (SWR) equilibria of AG(ϵ,c). Finally, let Γc be the set all strategies γ that are limits of strategies in Γϵc as the noise vanishes: γ∈Γc iff ∃{ϵi} and {γϵi} with γϵi∈Γϵic, {ϵi}→0, and {γϵi}→γ. Define Ψc similarly. Notice that Γc and Ψc are subsets of the set of equilibria of AG(0,c). We can then prove the following result.
Proposition 5There exists a thresholda∗(c)such that
In every member ofΓcandΨc, investment occurs for alla<a∗(c).
vH−a∗(c)→0 (at least as fast asc) asc→0. Thus, asc→0, every member ofΓcandΨcinvolves coordination on investment if and only if it is efficient.
The intuition of the proof will be sketched here; the formal details are in the appendix. The discussion below should also shed some light on why it is difficult to extend the result to non-simple strategies with multiple waiting regions (or provide a counter-example). The basic idea is to begin with an arbitrary strategy
s and “shuffle” its constituent regions to generate a different strategy
s′ that has the form of either the pessimistic example (
I−WN−N) or the partial coordination example (
I−WI−WN−N). Then, indifference relations that apply in an equilibrium best response to
s can be mapped into preference inequalities that apply to a best response to
s′. The advantage of this mapping is that the derived inequalities will be substantially easier to characterize than the original indifference relations.
This approach is illustrated for a simple strategy in Figure 5. Part (a) depicts the equilibrium strategy s. We draw attention to two points: aI is the highest signal below which I is always played, and aN is the lowest signal for which N is ever played. Since s represents a symmetric equilibrium, each agent must be indifferent between playing I and WI with signal aI, and indifferent between N and WN with signal aN, if his opponent is playing s. Part (b) shows a tweaked version s′ of s in which the entire strategy to the right of aN is replaced with N. Increasing the chance of facing N tends to make WI more attractive relative to I, so if WI I when facing s at aI, then we have WI≽I when facing s′ at aI. Similarly, the shift from s to s′ can only reduce the chance of facing I at aN, so if N WN when facing s at aN, then N≥WN when facing s′ at aN as well. We can now proceed with these two preference inequalities as for the pessimistic equilibrium of Section 3.
Figure 6 shows the procedure for a SWR strategy. In this case, s′ involves shuffling the interval (aI,aN) to shift all play of I (WN) to the left (right). Strategy I does equally well as WI against either I or WI but elicits full investment against WN, while WI does not. At aI, an agent faces WN less often under s′ than under s, so if he is indifferent between I and WI against s, he must prefer WI at aI when facing s′.
Now consider aN. WN outperforms N only when the opponent chooses I; otherwise both earn 0. An agent with signal aN faces I less often under s′ than under s, so in the best response to s′, N≥WN at aN. We are left in a situation similar to the partial coordination equilibrium above. The first two constraints will imply that if vH−aI fails to converge to 0 as c→0, then the WI region must expand to fill the entire interval (aI,aN). But this would mean that the chance of facing N after both agents have waited must go to 0. By appealing to a third constraint, that WN is preferred to WI at aN, we arrive at a contradiction. This is also where the difficulty with extending the proof to non-simple, multiple waiting region strategies lies. If WN were played on a region slightly above aN, then the probability of facing N after both agents wait need not go to zero, even if WI expands to fill (aI,aN). Thus, the contradiction needed to rule out the possibility that vH−aI is positive in the limit cannot be established.
5 Discussion
We have shown that for a large class of equilibria, strategic uncertainty interacts with the possibility of delay to restrict outcomes in a sensible way: investment occurs whenever it is efficient and worth waiting for. Both strategic uncertainty and delay are necessary for this result – the former provides an incentive for costly delay, and the latter provides a way to break up inefficient equilibria. Furthermore, the form of the strategic uncertainty is crucial as well; the correlation in the agents’ information and the “boundary conditions” imposed by the dominance of each action in an extreme region mean that even as ϵ goes to zero, there is yet some agent in some state of the world whose uncertainty about his opponent’s action is large. It is the unraveling started by this agent’s decision to wait that generates the result.
Endogenous timing in coordination games has been well explored by Chamley and Gale (1994) and Gale (1995) among others, but the study of how endogenous timing and global game-style strategic uncertainty interact is more recent. A key implication of delay in our model is the possibility of two types of learning: learning about the fundamental (because the other player’s action partially reveals his signal about a), and learning the action the other player is committed to. The latter, resolution of strategic uncertainty, is most critical to our results, particularly as noise vanishes. Dynamic coordination games (e.g. Chamley (2003)), and particularly, dynamic global games (e.g. Angeletos, Hellwig, and Pavan (2007)) often feature the first type of learning. Both of these involve a continuum of agents who must (repeatedly) try to coordinate on a risky attack; choosing not to attack in a period is in some sense a choice to wait or delay. In both cases, agents learn more about the threshold for a successful attack over time (publicly in Chamley, privately in Angeletos et al.). However, there is no scope for agents to resolve strategic uncertainty in either case, even if others’ past actions were observed, because actions are not “sticky” – actions are chosen anew each period, so past actions do not constrain current ones. Both models also admit multiple equilibria, but for different reasons than in our setting. For example, in Angeletos et al., multiplicity can arise because of the way that privately updated beliefs can shift over time, whereas in our setting multiplicity arises because common knowledge of bounds on others’ beliefs can re-emerge after both agents wait. In Angeletos et al., the option value of “delaying” one’s attack is principally the value of acquiring additional information about the fundamental before acting. Steiner (2008) adds an intertemporal payoff link: an agent’s capacity to profit from the risky investment depends on a stock variable – akin to liquid resources – that can be depleted by past failures to coordinate. Consequently, the main value of delay lies in conserving resources that might be better deployed at a future date. Thus, incentives to delay are greater when future prospects look more promising; in conjunction with the interaction of agents’ noisy information about fundamentals, this can deliver cycles of coordination and delay in equilibrium. In contrast, in our model, the option value of delay is mainly the prospect of learning which action one’s opponent has committed to.
Closer to our setting are models of of global games in which each agent faces a coordination-sensitive investment decision just once, not repeatedly, but perhaps has the option to either delay investing or to exit from an unfavorable investment. Araujo and Guimaraes (2015) investigate delay in an overlapping generations setting where current agents undertake a costly investment to benefit the older generation in the expectation that the next generation will reward them in similar fashion. The cost of that investment follows a random walk, and a current agent can choose when to invest, leading to a threshold rule in which agents delay investing until the cost is cheap enough. There are structural similarities to a one-shot global game between the current and subsequent agent, with drift in the cost playing the role of the correlated beliefs about a common parameter. Extending this analogy, the authors show that as the “noise” vanishes (i.e., as the investment cost converges to a constant), coordination on investment occurs only for costs low enough that it is the risk-dominant equilibrium, even with the possibility of delay. While their model differs from ours in many respects, the starkly different conclusion about the impact of delay relates once again to strategic uncertainty. In the OLG setting, delay, and more broadly, the sequentiality of actions, cannot help in resolving strategic uncertainty – observing the action of agent n−1 is not particularly helpful to agent n, as it is agent n+1 that she must hope to coordinate with.
The papers most closely related to this one are Ková⌣ and Steiner (2013), Dasgupta (2007), and Dasgupta, Stewart, and Steiner (2012). Both Dasgupta and Ková⌣ and Steiner focus on a continuum of agents and treat a single coordination-sensitive investment decision which may be delayed until or (for Ková⌣ and Steiner only) reversed after the arrival of new information. In Dasgupta, that new information relates to the fraction of agents who invested immediately. In the continuum setting, observing this fraction would reveal the fundamental perfectly, permitting multiple equilibria to re-emerge in the continuation among waiters; to avoid this, waiters are assumed to observe the number of early investors with noise – as strategies may be inverted, this amounts to observing a new noisy private signal about the fundamental, turning the continuation back into a global game. This is one point of contrast with my paper – in a large market it might be implausible for agents to have precise information about how others have acted, whereas with a small number of agents, precise information is not implausible, as we argued in the introduction. This leads us to permit past actions to be observed in our two-agent setting and to handle the resulting multiplicity by focusing on characteristics of the set of continuation equilibria. A second point of contrast is that Dasgupta specializes to normally distributed noise and focuses on monotonic equilibria, allowing him to derive extensive comparative statics results, while we look for weaker results on a larger class of equilibria and for arbitrary specifications of noise. Dasgupta finds a more limited welfare improvement from the delay option than we do: coordination on investment is achieved only for a subset of the range of fundamental values where it would be efficient, and efficient investment is most likely for intermediate rather than for low delay costs. These differences appear to be driven in large part by the number of agents who are trying to coordinate. With a continuum of agents, Dasgupta identifies a free-riding effect – if delay costs are low, too many agents may prefer to wait for better information rather than help supply it by stepping forward and leading with the efficient action. With two agents, leadership involves less of a risk – as long as the other agent has waited, he will follow suit – and consequently, lower delay costs always improve welfare.
Kováč and Steiner add a second stage in which agents may undo either the decision to invest (i.e., they can exit) or the decision not to invest (i.e., delayed investment is allowed) after observing a new private signal about the fundamental. They also study a rich family of payoff functions over both stages, allowing coordination payoffs to depend on contemporaneous investment by others and spillovers from future or past investment. Our model is closest to their setting with the option to delay investment (a reversible not-invest decision) and no spillovers (as all payoffs depend only on the final number of investors). In this case, Kováč and Steiner present an irrelevance result: the option to delay investing has no effect on the equilibrium chances of successful coordination. Setting aside other differences between the models, the difference in conclusions again appears most attributable to the number of agents. The new information that arrives in their model may be interpreted as a noisy signal about the mass of first-stage investors, but because each individual has a negligible effect on this mass an individual agent has no incentive to invest early in the hope of influencing those who delay. In contrast, with a small number of agents, each agent can make investing discretely more attractive for delayers by acting early; in particular, with two agents, either can make investing dominant for the other by acting early.
Finally, Dasgupta, Stewart, and Steiner (2012) also consider a setting with two agents, and an irreversible coordination-sensitive investment decision that can be delayed. There are several key differences: a decision can be delayed up to T times (with the sharpest results coming as T→∞), the other player’s actions are not observed, in particular, neither player ever learns for sure that the other has committed to the project, and new private signals about the fundamental arrive each period. They find that, notwithstanding the non-observability of actions, the option of arbitrarily long delay erodes strategic uncertainty and permits coordination on investment whenever it is undominated. The logic is roughly that when the fundamental is good enough for investment to be undominated, a player given long enough to learn will eventually become both eq. (1) quite confident that this is so, and eq. (2) confident that the other player will at some point be even more bullish on investing than he is right now. This provides the kernel for a contagion argument (similar to the usual one supporting the risk-dominant equilibrium) that investment will occur. In both their model and ours, it is the resolution of strategic uncertainty that is the key to effective coordination; our paper illustrates that this need not require the possibility of arbitrarily long delay if actions are observable.
We conclude with a few comments on robustness. While it is important for our results that a delayer can observe how a non-delayer acted, the results can remain valid if this observation is a bit noisy. For example, let a non-delayer’s action be observed by a delayer with probability 1−ϵ; with probability ϵ the delayer sees nothing and concludes that the other player has also waited. Alternatively, suppose the non-delayer’s action is always observed, but is perceived incorrectly with probability ϵ. It is not hard to confirm that either of these perturbations will have has a small effect on the equilibria of Sections 3.1 and 3.2 and that our limiting results for small delay costs go through unchanged. We conjecture that small changes in (or new information about) the investment cost at the second stage would have a similarly modest effect – the impact of the complete resolution of strategic uncertainty (if the other player has acted already) should tend to swamp such changes. Extensions to more than two players are less clear. Typically, commitment by several players (rather than just one) will be necessary to make following suit a dominant strategy for the others. This should tend to make the path to efficient coordination less immediate, but it may still be possible if there are several rounds of delay – investigating this question is left to future work.
More broadly, our results echo the note of caution that others have raised about refinements (such as the global games approach) that aim for sharp, robust behavioral predictions by perturbing the information structure of agents.
Weinstein and Yildiz (2007) give broad conditions under which any rationalizable outcome of the underlying game can be made the (robustly) unique equilibrium outcome under some perturbation of beliefs. In this sense, the merits of a particular equilibrium selection rest squarely on how convincing we find the belief structure that delivers that selection. Our paper illustrates that the implications of a particular belief structure may also be quite sensitive of other features of the strategic environment – in this case, flexibility (or its lack) in the timing of actions. Of course our results are subject to the same critique; while we have suggested above a few directions in which they are likely to be robust, further work is needed to determine how generally they may apply.
Appendix
Verification of the claims leading to Proposition 1
Claim 1: πI(a1) is strictly decreasing.
Observe that πI(a1)=vH−ϕ((aˉ−a1)/ϵ)Δv−E(a˜|a1). Both the second and third terms are strictly decreasing in a1.
Claim 2: There exists aˉ<vH such that πW(a1) is strictly decreasing and crosses 0 at aˉ.
For the first part, fix an arbitrary aˉ<vH. Then πW(a1)=(1−ϕ((aˉ−a1)/ϵ))(vH−E(a˜|a1;a2<aˉ))−c, where both terms in parentheses are positive and strictly decreasing in a1 for a1<vH. For the second part, substitute πW(aˉ)=12(vH−E(a˜|a1=aˉ;a2<aˉ))−c. At aˉ=vH, this expression is less than 12(vH−(vH−ϵ))−c=ϵ/2−c<0. At aˉ=vL, πW(aˉ) is greater than 12(vH−vL)−c > 0. By continuity, πW (ā) =0 for some aˉ in (vL,vH).
Claim 3: πI(a1)−πW(a1) is strictly decreasing and crosses 0 at a, with vL<a_<aˉ.
πI(a1)−πW(a1)=c−ϕ((aˉ−a1)/ϵ)(E(a˜|a1;a2>aˉ)−vL). The components of the second term are both positive (the latter for a1>vL) and increasing, so πI(a1)−πW(a1) is strictly decreasing for a1>vL. Furthermore,
πI(vL)−πW(vL)>c−ϕ((aˉ−vL)/ϵ)((vL+ϵ−vL)>c−ϵ/2>0
If we can show πI(aˉ)−πW(aˉ)<0, then we are done. Toward this end, note that
πI(aˉ)−πW(aˉ)=c−(E(a˜|a1=aˉ;a2>aˉ)−vL)/2=(vH+vL)/2−(E(a˜|a1=aˉ;a2>aˉ)+E(a˜|a1=aˉ;a2<aˉ))/2=vˉ−aˉ
The second line follows by adding the identity πW(aˉ)=0. Furthermore, πW(aˉ)=0>(vH−aˉ)/2−c=(Δv2+vˉ−aˉ)/2−c=(Δv/4−c)+(vˉ−aˉ). By assumption, the first term is positive, so we must have aˉ>vˉ. But then πI(aˉ)−πW(aˉ)<0 as claimed, so there must be some a such that πI(a_)−πW(a_)=0 and vL<a_<aˉ.
Proof of Proposition 4
Step 1: (3) and (4) have a unique solution.
Rewrite the equations in terms of kˉ and a_, so (3) becomes
ϕ(kˉ)(vH−E(a˜|a1=a_+kˉϵ,a2<a_))=c
Note that the left-hand side of eq. (4) is increasing in a_, and for a fixed kˉ can be made larger than c (for ϕ(kˉ)>c/Δv) by taking a_ close to vH and smaller than c by taking a_ close to vL. Write am(kˉ) for the unique a_ that satisfies eq. (4), given kˉ. Because the left-hand side of eq. (4) is decreasing in kˉ, am(kˉ) is increasing.
Next, note that the left-hand side of eq. (3) is decreasing in a_. For a fixed kˉ, it will be larger than c (again, as long as ϕ(kˉ)>c/Δv) for a_ near vL and smaller than c for a_ near vH. Write an(kˉ) for the a_ that solves (3), given kˉ. LHS(3) decreasing in kˉ implies an(kˉ) decreasing. Furthermore, am(0)≈vL+2c<an(0)≈vH−2c, and am(ϕ−1(c/Δv))≈vH>an(ϕ−1(c/Δv))≈vL, so there is a unique kˉ such that am(kˉ)=an(kˉ).
Step 2: (a_,aˉ) is a best response to (a_,aˉ)
Suppose that 2 plays (a_,aˉ). Clearly I is a best response for 1 for any signals that rule out facing N, i.e., for a1≤aˉ−2ϵ. For a1=a_−x>aˉ−2ϵ, the expected payoff difference between I and W is
Pr(W)(vH−E(a˜|a_−x,a2∈[a_,aˉ]))−Pr(N)(E(a˜|a_−x,a2>aˉ)−vL)+c(ϕ(x)−ϕ(x+kˉ))(vH−E(a˜|a_−x,a2∈[a_,aˉ]))−ϕ(x+kˉ)(E(a˜|a_−x,a2>aˉ)−vL)+cϕ(x)(vH−a′)−ϕ(x+kˉ)(Δv+Δa)+c
where a′=E(a˜|a_−x,a2∈[a_,aˉ]) and Δa=E(a˜|a_−x,a2>aˉ)−E(a˜|a_−x,a2∈[a_,aˉ]). Both a′ and Δa are decreasing in x; for Δa, this is a consequence of logconcavity of G. Furthermore, logconcavity of ϕ implies that ϕ(x)/ϕ(x+kˉ) is increasing in x. Suppose that 1 were indifferent between I and W for some positive x∗. Then ϕ(x)(vH−a′)−ϕ(x+kˉ)(Δv+Δa)<0 on a neighborhood of x∗. But then the payoff difference can be written
ϕ(x+kˉ)(ϕ(x)ϕ(x+kˉ)(vH−a′)−(Δv+Δa))+c
The term in parentheses is negative near x∗ and increasing (becoming less negative) in x. It is multiplied by a term that is positive and decreasing in x, so the entire expression is increasing in x. That is, I(W) is strictly preferred for signals in a neighborhood below (above) a_−x∗. This is true for any indifference point x∗≥0, including x∗=0, so I must be strictly preferred for all signals below a_. (Otherwise the expected payoff difference would have to cross 0 with positive slope for some signal below a_, a contradiction.) This argument extends directly to show that W is strictly preferred to I for signals above a_.
As for the upper threshold, N is a clear best response for signals large enough that there is no hope (or fear) of facing I. The relative payoff to W (vs. N) is increasing in the probability of facing I, which in turn increases as a1 decreases, so aˉ is unique, and N(W) is strictly preferred for higher (lower) signals.
Proof of Proposition 5
Let aI be the boundary below which I is always played and let aN be the boundary below which N is never played, that is, aI=sup{aI′:I is played at a∀a≤aI′} and aN=sup{aN′:N is never played at a∀a≤aN′}. Suppose that aI=aN. Then continuity of payoffs requires that I and N both have an expected payoff of 0 at aI. A deviation to waiting at aI has payoff at least 12(vH−E(a˜|a1=aI,a2<aI))−c>12(vH−aI)−c(because with probability 1/2 one’s opponent has a lower signal for which I is always played). If aI<vH−2c, then this last term is positive, and waiting is strictly preferred at aI, and by continuity, in a neighborhood of aI, contradicting the definitions of aI and aN. Thus it must be that aI≥vH−2c≥vH−(c2+2cΔv−c) (for all c<Δv/4), and so the proposition is satisfied.
Suppose instead that aI<aN. We will establish a bound for equilibria with pessimistic conjectures about play after (W,W) and then show that the bound can only be tighter for other conjectures. We will write Pr(R|aR′) for the probability that 2 plays R conditional on a1=aR′, R,R′∈{I,N,W}. There are two cases to consider, depending on whether I is ever played on the interior of [aI,aN].
Case 1: Only W is played on [aI,aN]
Then 1 must be indifferent between I and W at aI, so we have
Pr(N|aI)(E(a˜|aI,2 plays N)−vL)−Pr(W|aI)(vH−E(a˜|aI,2 plays W))=c
The expectations of a˜ must lie within ϵ of aI, so
Pr(N|aI)(aI+ϵ−vL)>Pr(W|aI)(vH−(aI+ϵ))+c
Pr(N | aI)Δv>(1−Pr(I | aI))(vH−(aI+ε))+c>(1−Pr(I | aI))(vH−aI)+c−ε
Pr(N|aI)Δv+Pr(I and a2>aN|aI)(vH−aI)>(1−Pr(Iand a2<aN|aI))(vH−aI)+c−ϵ
(Pr(N|aI)Δv+Pr(I and a2>aN|aI))Δv>(1−Pr(Iand a2<aN|aI))(vH−aI)+c−ϵ
Pr(a2>aN|aI)Δv>(1−Pr(Iand a2<aN|aI))(vH−aI)+c−ϵ
Pr(a2>aN|aI)Δv>(1−Pr(a2<aI|aI))(vH−aI)+c−ϵ
Pr(a2>aN|aI)Δv>12(vH−aI)+c−ϵ
ϕ(kˉ)>(vH−aI)/2+c−ϵΔv
where kˉ=(aN−aI)/ϵ. The fact that Pr(a2>aN|aI)>0 establishes that kˉ<2.
Next, 1 must be indifferent between N and W at aN, yielding
Pr(I|aN)(vH−E(a˜|aN,2playsI))=c
Pr(I|aN)(vH−aN−ϵ)<c
Pr(I and a2<aI|aN)(vH−aN−ϵ)<c
ϕ(kˉ)(vH−aN−ϵ)<c
ϕ(kˉ)(vH−aI−3ϵ)<c
ϕ(kˉ)<cvH−aI−3ϵ
Together, these imply
x/2+c−ϵΔv<cx−3ϵ
letting x=vH−aI. We will return to this inequality after addressing Case 2.
Case 2: I is played somewhere on the interior of [aI,aN]
The logic of Case 1 hinged on two inequalities. In the latter, an agent who plays N at aN must expect to see I sufficiently rarely. When I is never played on [aI,aN] but always played below aI, this bounds kˉ below. If I is also sometimes played on [aI,aN], then in principle, a tighter lower bound on kˉ could be applied. On the other hand, in the former inequality, an agent who plays W at aI must expect to see N sufficiently often relative toW, putting an upper bound on kˉ relative to how often W is seen. When I is played somewhere on [aI,aN] it displaces W, so this upper bound must become looser. The difficulty in Case 2 is in showing that the tightening upper bound matters more than the loosening lower bound.
Suppose that σ is the measure of signals in the interval [aI,aN] for which I is played. Consider an alternative profile in which I is played everywhere on [aI,aI+σ] and W is played everywhere on [aI+σ,aN]. That is, all play of I is squeezed to the left end of the interval. The strategy of the proof will be to show that the inequalities that must hold for the true profile imply a set of inequalities for this alternative profile that can be treated in a way similar to Case 1.
First consider indifference at aN. Just as before, we have
Pr(I|aN)(vH−aN−ϵ)<c
(Pr(I and a2∈[aI,aN]|aN)+Pr(I and a2<aI|aN))(vH−aN−ϵ)<c
Now imagine squeezing all play of I to the left. Because the conditional distribution of a2 is log-concave, the probability that a2 lies in [aI,aI+σ] conditional on a1=aN is smaller than the probability that a2 lies in any other measure σ subset of [aI,aN] conditional on a1=aN. Thus we have Pr(a2∈[aI,aI+σ]|aN)< Pr(I and a2∈[aI,aN]|aN), giving us
[6]Pr(a2<aI+σ|aN)(vH−aN−ϵ)<cϕ(kˉ−σ/ϵ)(vH−aN−ϵ)<cϕ(kˉ−σ/ϵ)<cvH−aI−3ϵ
Now consider aI. Picking up from the sixth line above,
Pr(a2>aN|aI)Δv>(1−Pr(Iand a2<aN|aI))(vH−aI)+c−ϵ
Pr(a2>aN|aI)Δv>12−Pr(Iand a2∈[aI,aN]|aI)(vH−aI)+c−ϵ
Once again, imagine that all play of I occurs in [aI,aI+σ]. Again by log-concavity of G, the probability that a2 lies in [aI,aI+σ], conditional this time on a1=aI, is greater than the probability that a2 lies in any other measure σ subset of [aI,aN]. Consequently, Pr(I and a2∈[aI,aN]|aI)<Pr(a2∈[aI,aI+σ]|aI), so
[7]Pr(a2>aN|aI)Δv >12−Pr(a2∈[aI,aI+σ]|aI)(vH−aI)+c−ϵϕ(kˉ)Δv >ϕ(σ/ϵ)(vH−aI)+c−ϵϕ(kˉ)ϕ(σ/ϵ) >vH−aIΔv+c−ϵϕ(σ/ϵ)Δvϕ(kˉ)ϕ(σ/ϵ) >vH−aIΔv+2(c−ϵ)Δv
where the last step follows because ϕ(σ/ϵ)<1/2 for σ>0. As in Case 1, we would like to compare lines (6) and (6); logconcavity of G permits this.
Lemma 1Foru,v>0, ϕ(u)ϕ(v)≥ϕ(u+v)/2.
ProofWrite ψ(x)=lnϕ(x). Because ψ is concave (by logconcavity of G, which implies logconcavity of ϕ), we have ψ(0)+ψ(u+v)≤ψ(0+u)+ψ((u+v)−u)=ψ(u)+ψ(v). Substituting ϕ(0)=1/2 into this equation yields the result. ■
Applying this lemma to (6) and (6), we arrive at
cvH−aI−3ϵ>vH−aI2Δv+c−ϵΔv
just as in Case 1. Solving the resulting quadratic equation for x=vH−aI yields
x<−c+52ϵ+c2+cϵ+14ϵ2+2cΔv
Taking limits as ϵ goes to 0 establishes the result that was claimed.
Finally, consider an equilibrium in which an arbitrary combination of I, WI, and WN are played on [aI,aN], where Ws refers to waiting and playing s if the other agent also waits. At aN, N is at least weakly preferred to the more profitable of WI and WN, so we certainly have N≿WN. At aI, the better of WI and WN is weakly preferred to I. If this is WN, we have WN≿I at aI and N≿WN at aN, and we can proceed just as in Case 2 above. Suppose instead that I WI≿WN at aI. Then we proceed with the two conditions WI≿I at aI and N≿WN at aN. First note that, as before, if N is a better response than WN to the true equilibrium strategy at aN, it will also be a better response to the strategy in which N is always played to the right of aN (but which otherwise corresponds with the equilibrium strategy). This is also true for WI≿I at aI. Suppose that on the interval [aI,aN], I is played for a measure σϵ of signals and WI is played for a measure γϵ of signals. Similarly to Case 2, we can construct an alternative profile in which all play of I is shifted to the left end of [aI,aN] and all play of WN is shifted to the right end of the interval. WI and I have different outcomes only when N or WN are faced. At aI, this shift doesn’t affect the chance of facing N and reduces the chance of facing WN (the situation in which I does better than WI), so the reply preference WI≿I at aI continues to hold. WN and N have different outcomes only when I is faced (in which case WN is preferred to N). This shift reduces the chance of facing I at aN, so the reply preference N≿WN at aN also continues to hold. To summarize, we have two conditions on best replies to the adjusted profile in which I is played below aI+σ, N is played above aN, WI is played on [aI+σϵ,aI+σϵ+γϵ], and WN is played on [aI+σϵ+γϵ,aN]. As above, the condition that N≿WN at aN leads to the inequality
ϕ(kˉ−σ)<cvH−aI−3ϵ
The condition WI≿I at aI leads to the inequality
ϕ(kˉ)(aI−vL+Δv+ϵ)>ϕ(σ+γ)Δv+c
Thus, as ϵ→0, we have
[8]ϕ(kˉ−σ)<cvH−aI
[9]ϕ(kˉ)(aI−vL+Δv)>ϕ(σ+γ)Δv+c
There are two subcases to consider. Suppose first that in the limit of the ϵ equilibria, WN is preferred to WI at aN. Then WN will still be preferred under the shifted profile (as it makes facing WI less likely). But this means that the probability of facing WN conditional on aN and both agents waiting must be greater than p∗, where p∗ is defined by
(1−p∗)(vH−aI)+p∗(vL−aI)= 0p∗= vH−aIΔv
Under the shifted profile, this probability is just
p′=ϕ(0)−ϕ(kˉ−σ−γ)ϕ(0)−ϕ(kˉ−σ)
Now we combine the first two ϵ→0 conditions with p′≥p∗, and suppose toward a contradiction that vH−aI were to converge to 0 more slowly that at rate c as c→0. From (8), ϕ(kˉ−σ) would have to go to 0 at a rate faster than c. But then because ϕ(kˉ)<ϕ(kˉ−σ), ϕ(kˉ)→0 faster than c as well. Then (9) means that ϕ(σ+γ)→0 faster than c. Together these mean that kˉ→2 and σ+γ→2, so ϕ(kˉ−σ−γ)→ϕ(0)=1/2. How quick is this convergence? For c small, so kˉ and σ+γ near 2, ϕ(kˉ−σ−γ)→ϕ(0) faster than cϕ′(kˉ−σ−γ)ϕ′(σ+γ) which is greater than c because ϕ is steeper near 0 than near its tails (because it is single-peaked). Altogether, this means that p′→0 faster than c, but then p′≥p∗ means that vH−aI→0 faster than c, a contradiction.
Now consider the second subcase, in which WI is preferred to WN in the limit of the ϵ equilibria. Then the probability of facing WI, conditional on aN and both agents waiting must be greater than 1−p∗. Then the condition that N≿WI at aN gives us
ϕ(kˉ−σ)+Pr(faceWI)<cvH−aI
and Pr(face WI)>Pr(face W)(1−p∗). Furthermore, Pr(face W)<1/2−ϕ(kˉ−σ), so we have
12−vH−aIΔv12−ϕ(kˉ−σ)<cvH−aI
In this case, vH−aI must go to 0 at least at rate c. If it did not, the right hand side would go to 0, so the left hand side would have to do so as well. This would only be possible with ϕ(kˉ−σ)→0 and aI→vL. But if aI (and hence aN) goes to vL, then choosing I would be strictly better (with a payoff at least Δv/2) at aN than N, another contradiction. Thus, in every case, the limiting aI (as ϵ→0) goes to vH as the waiting cost vanishes at rate c or faster.
