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Article Excerpt
Summary. We introduce strategic waiting in a global game setting with irreversible investment. Players can wait in order to make a better informed decision. We allow for cohort effects, which arise endogenously in technology adoption problems with positive contemporaneous network effects. Formally, cohort effects lead to intra-period network effects being greater than inter-period network effects. Depending on the nature of the cohort effects, our game may or may not satisfy dynamic increasing differences. If it does, our model has a unique rationalizable outcome. Otherwise, multiple equilibria may exist as players want to invest at the same point in time others do.
Keywords and Phrases: Global game, Strategic waiting, Coordination, Strategic complementarities, Period-specific network effects, Equilibrium selection.
JEL Classification Numbers: C72, C73, D82, D83.
1 Introduction
Often the optimal action of an economic agent is complementary to the actions undertaken by other agents. For example, a consumer's payoff from buying computer software is typically increasing in the number of other consumers who use that software. Or, think of a consumer who decides to buy a durable consumption good such as a car. As more consumers buy this brand of car, more repair shops will have the know-how and spare parts to repair the car quickly. (3) Models of situations in which the agents' optimal actions are complementary to each other are often plagued by multiple equilibria with self-fullfilling beliefs: If a player expects the other players to buy the software, then it is in her best interest to buy it as well. If a player expects the other players not to acquire the software, she wants to refrain from buying. This multiplicity is annoying from an economic policy point of view. Without an adequate theory of equilibrium selection, one cannot use these theories to predict the market outcome. How then does one judge, for example, whether policies to subsidize/tax the adoption of information technology should be implemented? How does one predict the market power of firms who sell their products in markets with network externalities?
For two-player coordination games, Carlsson and van Damme [4], henceforth CvD, developed an equilibrium selection theory, which Morris and Shin [23] extend to a continuum of players. CvD assume that the agents' payoffs depend on the action chosen by the other agent in the economy and some unknown state of the world [theta]. Agents receive different signals about [theta], which generate beliefs about the state of the world and a hierarchy of higher order beliefs. CvD refer to their model as a global game and give conditions under which the equilibrium is unique. (4)
Thus, the global game framework enables researchers to predict behavior in coordination games. It has been applied to a wide variety of contexts within a static framework. (5) In reality, however, many economic coordination problems are essentially dynamic. Players can always postpone their investment decisions in order to make a better informed decision. This paper investigates conditions under which the global game approach can be extended to model dynamic technology adoption problems.
To address this question, we build a dynamic global game. We consider a continuum of investors, who have the opportunity to engage in a risky investment project in either of two periods. Investments are irreversible. Payoffs depend positively on the realization of a random variable, which we refer to as the fundamental, and on the mass of investors. All players receive some noisy private information concerning the realization of the fundamental. For very high signals, it is a dominant strategy to invest immediately and for very low signals it is a dominant strategy not to invest. For intermediate signals, a player's optimal behavior depends on how other investors act. If a player decides to wait, she gets a more informative signal concerning the realization of the fundamental at the cost of foregone profits. We work with a flexible dynamic payoff structure in which a player's gain of investing not only depends on the total mass of players who invest, but also on when the other players invest. We say that our payoffs exhibit an early (late) mover cohort effect if the early (late) adopters enjoy more network benefits from the other early (late) adopters than from the late (early) ones. Four main results emerge from our analysis.
First, we show that cohort effects can arise endogenously in a dynamic set-up with contemporaneous network effects. We discuss three archetypical technology adoption problems. In the first, which we call "Fixed Lifespan" (FL), players decide to adopt a technology that becomes obsolete after two periods. In the interim period in which late movers have not invested yet, early movers are subject to a contemporaneous network effect that only depends on the mass of early movers. Since the technology of the early movers becomes obsolete while late movers are still using the technology, late movers benefit from a contemporaneous network effect in period 3 that depends only on the mass of late movers. This interpretation is thus characterized by an early and a late mover cohort effect. In the second interpretation, which we call "Joining a Nascent Club" (NC), players must decide whether to become member of a club. The more players who join the club, the greater its appeal. At time one the club's member base may grow in the future. At time two the club's member base has reached a "mature" level (and will remain constant in the future). For the same reason as above, this interpretation exhibits an early mover cohort effect. For late members, however, the network benefit in any period depends only on the total mass of members (regardless of when the other players joined the club). Hence, this interpretation is void of any late mover cohort effect. In the last interpretation, which we refer to as "Pledging to Invest" (PI), players commit whether or not to invest before the technology becomes available. As the technology becomes available to all players at the same point in time, there is neither an early nor a late mover cohort effect.
We next introduce a condition on the ex-post payoff function called dynamic increasing differences. Call both a change from not investing at all to investing at time two and a change from investing at time two to investing at time one, a move to a higher action. (6) Dynamic increasing differences implies that as a higher percentage of the population takes a higher action, it becomes weakly more profitable to take a higher action. For example, it requires that as more players invest late rather than not at all, it becomes weakly more profitable to invest at time one. Our second result shows that a technology adoption problem that exhibits contemporaneous increasing differences does not necessarily exhibit dynamic increasing differences. In particular, dynamic increasing differences requires there to be no late mover cohort effect and it is thus violated by the FL interpretation. The other interpretations, however, satisfy dynamic increasing differences.
Our third result proves that dynamic increasing differences imply the existence of a unique rationalizable outcome. We start by observing that active players who have a "very high" second-period signal always want to invest, (7) since they believe that the fundamental is so good that investing is profitable no matter what actions the other players choose. Now consider a player who has an "extremely high" first-period period signal so that she foresees that her second-period signal will be very high even if she gets bad news. She strictly benefits from investing immediately and saving the waiting costs if she expects no other player to invest.
Now consider a player who has a "high" but not an "extremely high" first-period signal. If she expects no other player to invest in either period, then she would also prefer to refrain from investing. As she possesses a flat prior concerning the realization of the fundamental, it is equally likely that the other players received a higher or lower signal than herself. Therefore, in equilibrium, she cannot expect that no one invests. As her signal is "high," her knowledge that all players with an extremely high signal invest at time one and that all active players with a very high signal invest at time two, induces her to invest at time one as well. Similarly, the knowledge that all players with an extremely high signal invest at time one and that all active players with a very high signal invest at time two, gives active players with a high (but not a very high) signal a strict incentive to invest at time two. This will, in turn, convince players with slightly less favorable signals to also invest, etc ... This process of iterative elimination of dominated strategies ends at some cutoff vector ([bar.k.sub.1], [bar.k.sub.2]). Mirroring the above argument, there is a critical cutoff vector ([k.bar.sub.1], [k.bar.sub.2]) such that a player refrains from investing in period t whenever she has a signal below [k.bar.sub.t]. We next observe that these cutoff vectors give rise to symmetric switching equilibria. In the final step, we exploit the nature of symmetric switching equilibria to show that if the ex-post payoff function satisfies dynamic increasing differences, then the extremal switching equilibria coincide, i.e. ([bar.k.sub.1], [bar.k.sub.2]) = ([k.bar.sub.1], [k.bar.sub.2]).
Fourth, we characterize symmetric switching equilibria for a wide range of parameter values. This enables us to illustrate why multiple equilibria can arise if dynamic increasing differences are violated. In essence, if dynamic increasing differences are violated, then players have an incentive to invest at the same point in time at which other players do. If this incentive is strong enough, it gives rise to self-fullfilling expectations according to which some players invest at time two if and only if they anticipate other players to do the same.
This is not the only paper to introduce dynamic elements in a global game. Chamley [5] studies a dynamic global game in which there is uncertainty about the distribution of investment costs. The distribution of investment costs evolves stochastically through time. Players use the observed previous behavior to update their beliefs about the state of the world. If there is sufficient heterogeneity in the population, each period can be analyzed as a static global game and the equilibrium is unique. (8) Contrary to our paper, there is a new population of players in every period. Thus, players cannot choose when to invest.
Morris and Shin [24] study the onset of currency crises using a dynamic global game in which the fundamental follows a Markov process. As long as there has been no successful attack, all players choose whether or not to attack in every period. In each period, the past realizations of the fundamental are common knowledge and players observe a private signal regarding its current realization. If the private signal is sufficiently precise, each period can be analyzed as a static global game and the model has a unique equilibrium. (9) In contrast to our model, investments are not irreversible.
Dasgupta [8] introduces strategic waiting in a global game with irreversible investment. Players can invest in two periods. If a player delays, she observes a noisy signal about the past economic activity at the cost of foregone profits. Dasgupta provides a condition under which his game is characterized by a unique equilibrium within the class of symmetric switching strategies. In his model players wait to benefit from social learning, while in our model players delay to obtain a more precise signal. Furthermore, we allow for cohort effects and do not restrict attention to symmetric switching strategies.
Burdzy et al. [3] investigate a complete-information dynamic model in which the state evolves stochastically through time. In each period, a continuum of players is randomly matched to play a 2x2 game with strategic complementarities. Under the assumptions that (i) in some states of the world playing one action is dominant while in others the other is dominant and that (ii) in each period a player has only a small chance of revising her action, they characterize the unique equilibrium. (10) A similar set-up is used by Frankel and Pauzner [14] to investigate a model of sectoral choice and by Oyama [27] to analyze economic fluctuations in less developed countries. In contrast to our paper, these papers require that only a small set of players can revise their action at any given point in time and they do not allow for strategic waiting.
Echenique [11] investigates the set of subgame perfect equilibria in extensive-form dynamic games with strategic complementarities. While his set-up differs from ours, he also observes that static strategic complementarities do not imply dynamic complementarities--although for a different...
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