UTILITY AND ENTROPY IN SOCIAL INTERACTIONS

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1 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS MARCIN PĘSKI Abstract. We immerse a single-agent Dynamic Discrete Choice model in a population of interacting agents. The main application is a strategic and dynamic model of many-to-many matching. We characterize equilibria as solutions to an optimization problem with an objective function that consists of a systematic utility term plus the entropy of the distribution over agent histories. All identified parameters can be described by closed-form functions of data. In particular, various joint surpluses or payoff externalities can be identified. We show that transferable and non-transferable versions of the model are directly related and one can move between the two versions with a simple re-scaling of parameter values. 1. Introduction Dynamic Discrete Choice (DDC) models study a single Bayesian and forward-looking agent who sequentially chooses actions. Each choice weighs immediate payoff gains with uncertain long-term consequences. The goal of this paper is to extend the basic model to situations in which decisions are made jointly with other, randomly-met agents. Examples include marriage or employment, either of which can occur only if the two participating sides simultaneously agree to it. The main application is to a strategic and dynamic model of one-to-one or many-to-many matching, such as marriage search, job search with multiple jobs and multiple workers, co-authorship or friendship networks, and club formation. The main idea is that from the perspective of a single agent who considers accepting the shared action, the choice made by the other participant can be treated as a random event that determines whether the action is available for the agent. For this reason, we introduce a version of the single-agent DDC with randomly available action Date: July 6, I am very grateful to Aloysius Siow and Ronald Wolthoff for discussion, as well as seminar participants at the University of Pittsburgh, Duke and Columbia for comments. Yiyang Wu and Jiaqi Zou for excellent research assistance. 1

2 2 MARCIN PĘSKI opportunities. In each period, an agent observes whether an action is available, and if so, decides whether or not to take it. Each action is associated with an i.i.d. payoff shock. A key assumption is that the payoff shock is exponentially distributed. In each period, the agent receives a systematic utility that may depend on past actions as well as some exogenous events that occurred to the agent (examples of which include major sickness or death of a spouse). Additionally, the agent who chooses to take the action, receives the associated payoff shock. We say that the behavior is interior if the agent always rejects actions that have the (lowest possible) payoff shock of value 0. The population model endogenizes the availability probabilities of shared action opportunities. The agents meet randomly and independently decide whether to accept or reject the opportunity. From the point of view of each agent, the probability that an action is made available is proportional to the average acceptance rate in the population. The proportionality constant is an exogenous parameter that can be interpreted as market frictions. The average acceptance rates are determined endogenously, in equilibrium. From the point of view of the population, acceptance decisions form a game with complementarities: if some agents are more likely to accept shared opportunities, other agents are more likely to face opportunities with attractively high payoff shocks, which, in turn, leads to increases in their own acceptance rates. Despite its complexity, the model is remarkably tractable with a simple characterization of expected payoffs, equilibrium and a closed-form identification. We obtain the following results: (1) As in a standard DDC model, each acceptance strategy induces a probability distribution π over lifetime histories. We show that the expected single-agent payoff can be represented as a function of π : υ π + ρ π + r π + H (π). (1) The first term is the discounted sum of the expected systematic utility. The second and the third terms are respective corrections for the exogenous and endogenous components of the availability probabilities of action opportunities. The last term is an approximate version of the entropy of π (it converges to the Shannon entropy in the continuous time limit of the model) and it measures the randomness of π. Optimal behavior maximizes the above expression. The first three terms push behavior towards a particular lifetime history with a

3 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 3 large systematic utility and that is likely to become available. This tendency is balanced by the last term, which ensures that the optimal behavior is more random. (2) The average population welfare is a function of the mass distribution m over population lifetime histories, and it is equal to υ m + ρ m H (m S ) + H (m). (2) The first, second and last term have a similar interpretation as the corresponding terms of (1). The third term replaces the endogenous component of the availability probabilities from (1). It is equal to the negative entropy of the marginal distribution over shared actions only. We interpret the third term as a measure of coordination gains that arise due to search complementarities. The more coordinated is the acceptance behavior, the more concentrated and the less random are the masses of shared actions. (3) We show that if equilibrium strategies are interior (and we provide a condition for this), an equilibrium mass m is a critical point of the functional υ m + ρ m H (m S ) + H (m) I m S (3) subject to some natural constraints. The sum of the first four terms is equal to the average welfare as in (2). The last term is equal to the weighted and discounted average mass of shared actions, and it measures the inefficiency that arises because agents do not internalize the utility loss of their partners when rejecting an opportunity. The constraints ensure that the masses of shared actions from sides of the market agree. There are three reasons why this result is significant. First, it simplifies the problem of finding an equilibrium by replacing a fixed-point problem with a computationally simpler optimization problem. Second, it leads to a straightforward welfare analysis. The last term of (2) suggests that the welfare of each equilibrium could be improved if agents can be induced to take more shared actions (perhaps through subsidies, with the subsidy value given by function I). Third, it allows us to precisely describe the limits of identification. (4) We describe reduced-form parameters that are functions of the fundamental parameters υ and ρ, and are identified from data through simple closed-form formulas. The reduced-form parameters are sufficient to compute (3) and to

4 4 MARCIN PĘSKI find an (interior) equilibrium. We show that nothing beyond the reduced-form parameters can be identified. More precise identification can be obtained in applications. For example, in a simple model of many-to-many matching, the individual utility is not identified, but the second-order properties of the individual utility (i.e., payoff externalities exerted by one connection on the payoffs from the other connections), as well as the joint payoff surplus from a connection (i.e., the sum of the payoff surpluses across participants) are identified. The latter observation generalizes the result from Choo (2015) to models of strategic and/or many-to-many matching. We use these results to discuss counterfactual predictions. Equilibrium data can be used to predict the new equilibrium under a combination of the two types of changes in parameters: additive change in the systematic utility (for instance, a subsidy per child), or a multiplicative change in the exogenous availability probabilities (for instance, a change in the speed of the market). (5) As an extension, we describe a version of the model with transferable utility (TU) and with transfers negotiated through take-it-or-leave-it bargaining. We show that the TU model is equivalent to the original non-transferable utility (NTU) model with modified parameters. In particular, all the above results (including identification) carry over to the TU model. Also, it is impossible to use choice data to distinguish between the NTU and TU models. (This latter fact coincides with the characterization of testable implications of stability in Echenique, Lee, Shum, and Yenmez (2013). A similar relation holds between the static cooperative NTU model of Dagsvik (2000) and the TU model of Choo and Siow (2006)). In the original model, we assume that all shared actions occur between two agents that who randomly and do not have any previous shared history. As another extension, we relax this assumption by allowing follow-up actions between agents that jointly participated in an earlier shared action. We use this extension to discuss a dynamic model of marriage with children. (6) We illustrate the model with special cases: one-to-one matching with a onedimensional parameter, a many-to-many matching, a dynamic model of marriage, and marriage with children. In the first case, we show that the matching is assortative if the joint utility is supermodular and we compare this result

5 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 5 with Shimer and Smith (2000). In the other cases, we show how an additional structure on the model leads to a clean interpretation of the identified information Literature review. The DDC models were first introduced in Wolpin (1984) and Rust (1987). For a recent survey, see Aguirregabiria and Mira (2010). Our approach differs in a few ways. First, while most of the DDC models allow for multiple choices, the choice in our model is always binary: to accept or reject an action. At the same time, note that because of the random availability of actions, the statistician observes a random choice from multiple options. The multiple choice of single-agent actions can be added at some expositional cost. The advantage of our approach is that it allows for a clean way of allowing for randomly available alternatives, which is essential for immersing the single-agent model in the population. Second, we assume that the rejection decision is not registered in the history of the agent, and that it cannot be distinguished from an action that was not available. This assumption reflects the nature of the data in which the absence of an action (for instance, marriage or employment) can be due to either (a) lack of opportunity, or (b) rejection either of, or (c) by the other participant, where the exact reason is typically not observed. Third, we assume that the payoff shocks are distributed exponentially. The role of this assumption is analogous to the extreme value type I error distributions in logit models. It allows us to make explicit computations and it makes the model tractable. Despite these differences, our single-agent results are related. The single-agent s identification is related to results from Hotz and Miller (1993) and Arcidiacono and Miller (2011). We further discuss the similarity of representation (1) to its analogue from the discrete choice literature below. A typical empirical work on matching relies on static models with cooperative solution concepts from classic matching theory (for NTU case, see Dagsvik (2000), Menzel (2015); for TU, see Choo and Siow (2006), Fox (2010) among others. Also, see Graham (2011) for a review). One difficulty of the cooperative approach is that a stable solution may fail to exist beyond the classic matching market of Gale and Shapley (1962). In order to guarantee existence, the literature typically either makes restrictive assumptions or studies approximate solutions. 1 Neither approach is fully satisfactory. 1 The existence of a stable solution under various type of assumptions on the preferences or structure of the model was studied by Kelso and Crawford (1982), Chung (2000), Hatfield and Milgrom

6 6 MARCIN PĘSKI Our model avoids this difficulty, as we use a non-cooperative approach for which an equilibrium exists due to standard, fixed-point arguments. A recent paper Choo (2015) considers a dynamic marriage search model. In each period, an agent decides whether to marry or stay single for the next period. There are two main differences. First, Choo (2015) clears the markets through the cooperative TU model of Choo and Siow (2006), while we are consistently strategic and our paper allows for both NTU and TU interpretations. Second, we allow for multiple matches for each agent, whereas Choo (2015) focuses on (monogamous) marriage. As a main result, Choo (2015) shows that the joint marriage utility is identified. Despite the differences, we show that that a single-match version of our model has a remarkably similar identification result. The utility plus entropy formulas appear at the intersection of different economic literatures. First, the representation (1) is similar to the well-known results in static logit models (see McFadden (1978)). Although we are not aware of analogous representation in dynamic models, we suspect that it is possible. Second, a connection between utility plus entropy and discrete choice (with a reverse in direction of the argument) has been established in the literature on rational inattention (Sims (1998), Sims (2003)). In that literature, an agent chooses a probability distribution over choices to maximize the utility plus entropy functional, where the entropy represents the cost of attention (or information). Matejka and McKay (2015) show that the solution of the rational inattention problem is similar in form to the random outcome of the static discrete choice model of McFadden (1978). Steiner, Stewart, and Matejka (2015) extend the analysis to a dynamic model, and they show that the solution resembles dynamic choice model with endogenous bias. Third, the same formula appears in various static cooperative matching models. Choo and Siow (2006) work with a transferable utility model with extreme value type I error term. (Differently than here, the error term is not entirely idiosyncratic as it is equal across all partners of the same type.) They show that the distribution of matches in stable outcome must satisfy a system of equations - it turns out that the equations are first order conditions to the welfare plus entropy maximization problem. An (2005). Echenique and Oviedo (2006) and Ostrovsky (2008) among others. Other papers establish the existence of approximate solutions (for instance, Tan (1991), Kojima, Pathak, and Roth (2013), Kojima, Pathak, and Roth (2013), Che, Kim, and Kojima (2015), Pęski (2017)).

7 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 7 analogous result for the NTU model of Dagsvik (2000) is established in Menzel (2015) under the assumption that the random utility terms satisfy a certain tail property. (Pęski (2017) extends the result further to roommate problem, which is a matching problem without Gale-Shapley algorithm). Galichon and Salanie (2012) generalize the TU insight of Choo and Siow (2006) to arbitrary random term distributions. In order to obtain their characterization, they replace Shannon entropy by its generalized version. The common thread among all these papers is that in each one of them, the individual agents preferences are formed in a version of the static discrete choice model. Our model is related to a substantial literature on search models in macroeconomics, labor or theory (for instance, Burdett and Coles (1997), Wright and Burdett (1998); for search with transferable utility, see Shimer and Smith (2000) and Atakan (2006)). In particular, we assumes random matching of candidates for shared actions. Additionally, the TU model of take-it-or-leave-it bargaining is very similar to an unemployment search literature where firms post wages, but search remains undirected (for instance, Burdett and Judd (1983), Albrecht and Axell (1984), or Burdett and Mortensen (1998)). An important difference is that, unlike the above papers, we are not restricted to the one-dimensional case, and our analysis does not rely on the assortativeness of the matching. However, in the special case of 1-1 matching with one-dimensional types, we show that, if utility is supermodular, equilibrium matching is assortative in the stochastic dominance sense. This result is opposite that of Shimer and Smith (2000). This is despite the fact that, like Shimer and Smith (2000) (and unlike Atakan (2006)), our paper also uses discounting as the source of frictions. This difference is the result of the random utility model and specific properties of the exponential distribution. (Burdett and Coles (1997) also obtain assortative matching under the assumption that assume that all men (and all women) have the same preferences.) There are few search papers with multi-dimensional preferences. Lauermann and Nöldeke (2014) consider a search marriage model with discounting-type of frictions. They show that, even if frictions disappear, the distributions over matches in some equilibria are not necessarily stable. Another exception is Coles and Francesconi (2016) which studies search with multidimensional types and random shocks. The authors prove the existence of equilibrium and use computational methods and calibration to estimate the differential impact of female features on match incentives.

8 8 MARCIN PĘSKI 1.2. Overview. Section 2 describes the model. In section 3, we analyze the singleagent problem. We describe a sufficient condition that ensures that best response strategies are always interior. Section 4 derives the welfare formula (2) and characterizes an equilibrium as a critical point of functional (3). Identification results can be found in Section 5. Section 6 describes an extension to a TU bargaining model. In Section 7, we discuss three examples. We show how additional structure leads to a cleaner interpretation of the identification results. Section 8 contains an extension to follow-up events. All proofs can be found in the Appendix. 2. Model We begin with a single-agent DDC model with randomly available actions. Next, we describe a population model, where the single-agent model is a building block and the arrival probabilities of some actions are endogenized Single-agent model. Time is discrete t Z. In each period t, an agent with history h observes a tuple (x, a, ε). Here, x is an exogenous event drawn from distribution P (. h) X over finite set X, a A { } is an action opportunity drawn from a distribution F (. h, x) (A { }), where A is a discrete set of actions, and represents inaction, and ε 0 is a payoff shock drawn i.i.d. from the exponential distribution ε G (ε) = e ε, where e is the base of the natural logarithm. If a, the agent decides whether to accept or reject the action opportunity. Probability distributions F are going to be endogenized in the population model. The distinction between x and a is that x occurs to the agent without his choice, and a occurs only upon acceptance. Examples of x include birth characteristics of the agent, major sickness, a death of the agent, or of the spouse. 2 Examples of a include going to school, taking up a job, or getting married. The distinction between (x, a) and ε is that ε is not observed by the econometrician. A (finite) history h = (b, x b, a b,..., x t, a t ) is a finite sequence of a birth date b followed by the pairs of an event x s and either an accepted action or inaction a s A { } for each period s {b,..., t}. Let b (h) = b denote the birth date and let t (h) = t 2 In the basic model, we do not explicitly assume that agents die. However, in a particular application, one of the exogenous events can be interpreted as death. examples. See Sections 7.2 and 7.3 for

9 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 9 denote the last period in h. For each t {b,..., t}, let h t = (b,..., a t ) be the t - period restriction of history h. History h is a continuation of h if h t(h) = h. Let H = Z s N (X (A { })) s denote the space of histories. For the purpose of the exposition, we often proceed as if event x is observed first, and then (a, ε) is drawn. We refer to (h, x) as an interim history. The total payoffs are equal to β s b (υ (b, x b, a b,..., x s, a s ) + 1 as ε s ), s b where β < 1 is the discount factor, υ (h) is the systematic utility received in period t (h) by an agent with history h, and ε s is the payoff shock associated with the (accepted) action opportunity in period s. Throughout the paper, we assume that υ is uniformly bounded. A (threshold) strategy is a mapping σ : H X A R + { }, with the interpretation that a is accepted at interim history (h, x) if ε σ (h, x, a). We allow for a possibility that some actions are never accepted. The restriction to threshold strategies is w.l.o.g. A strategy σ is interior if σ (h, x, a) > 0 for all h, x, a. We mention two important implications of the exponential distribution assumption. First, the probability of the action being accepted is equal to π σ (a h, x) = e σ(h,x,a). (4) Second, the expected value of the payoff shock conditional on the action being accepted is equal to E (ε ε σ (h, x, a)) = σ (h, x, a) + 1 = 1 log π σ (a h, x). (5) 2.2. Notation. Define infinite history h = (b, x b, a b, x b+1,...) as an infinite sequence starting with a birth date b = b (h) followed by event-action pairs. For each t b (h), let h t = (b, x b, a b,..., x t, a t ) be the (finite) history that is a restriction of h. Let H be the space of infinite histories. For each probability distribution π H over infinite histories, and for each (finite) history h H, let π (h) = π { h : h t(h) = h} be the probability that the infinite history h is a continuation of h. We define the π-probability π (h, x) of interim history (h, x) in an analogous way.

10 10 MARCIN PĘSKI Let Π H be the space of distributions π such that π (h, x) = P (x h) π (h) for each interim (h, x). (6) The constraint (6) ensures that exogenous events are chosen with probabilities P. We say that π Π are feasible probability distributions. (We postpone restrictions associated with action choice till later.) For each function f : H R, define f π = β t f (h t ) π (h) = β t(h) f (h) π (h). (7) t b h H h H This is a discounted expected sum of a sequence of payoffs given by function f, where the expectation is taken over distribution π. Below, we use π whenever we refer to a probability distribution over a history for a single agent, and we use m to denote the mass over histories in the population Population. There is a continuum of agents. The mass of agents born in period b is equal to P (b) 0, where b P (b) = 1. The state of the population is captured by a feasible mass m Π such that m (b) = P (b) for each b Z. (8) We divide actions depending on whether they involve one or more agents. For an instance of the former, a going to school is a single-agent action. For an instance of the latter, a taking a job is a shared action between two participants, an employer and an employee. Formally, let k (a) be the number of participants in a. The action is shared if k (a) > 1, otherwise it is single-agent. There is a correspondence S : A A such that S (a) = k (a), and S (a ) = S (a) for each a S (a). We refer to elements of S (a) as the associated actions taken by the other participants, or different roles in the shared action. 3 The assumption that shared actions have distinct roles adds generality. Also, it simplifies the statement of accounting identities (see equation (21) below). 4 3 In the employment example, a = "taking a job", a = "hiring a worker" and S (a) = S (a ) = {a, a }. 4 In some applications (e.g., co-authorship or friendship network, same-sex marriage), there are no natural roles for participants of shared actions. One way to say it is that participant roles do not affect utilities or availability rates. In order to fit such situations into the current setup, one may need to assign nominal roles.

11 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 11 We endogenize action opportunity probabilities F. Let σ be the acceptance strategy used in the population. (Because the best responses are unique in our model, it is w.l.o.g. to assume that each agent is using the same strategy.) Define P (. h, x) A as the (exogenously given) probability that the agent is available for action a. If a is single-agent, then, we take F (a h, x) := P (a h, x). If a is shared, we take F (a h, x) := P (a h, x) P (a h, x ) e σ(h,x,a ) m (h, x ). (9) a S(a)\{a} h :t(h )=t(h) 1 The above equation has the following interpretation. An agent becomes available for action a at rate P (a h, x). The agent meets available candidates for associated roles at a rate that depends on their numbers and availability. Once agents meet, each one separately decides whether to accept his or her role. The action occurs if it is accepted by all participants. Equation (9) describes the probability the original agent s decision is pivotal for accepting the action. Notice that the bracketed terms of (9) depend on the participant s role in the shared action a and the last period t in the history but nothing else. Let χ (a, t) := P (a h, x) e σ(h,x,a) m (h, x). (10) h,x:t(h )=t 1 be the (endogenous) probability of meeting an agent in period t who wants to participate in a. We refer to mapping χ as the endogenous acceptance probabilities. Define the probability that other participants accept their roles as ˆχ (a, t) := χ (a, t), (11) a S(a)\{a} when a is a shared action. To simplify notation, we take ˆχ (a, t) = 1 if a is a single-agent action or a =. The assumptions guarantee that χ (a, t), ˆχ (a, t) 1. The model is completely characterized by the flow utility υ, and probability distributions P (i.e., P (b), P (x h), P (a h, x) for all b, h, x, and a).

12 12 MARCIN PĘSKI 2.4. Equilibrium. An individual takes the behavior in the population, and in particular, the endogenous acceptance probabilities χ, as given. Let U (σ; b, χ) denote the expected payoff of the agent who is born in period b, who uses strategy σ, and who faces the arrival rates of actions equal to (9). A strategy σ is a best response of individual b given χ if it maximizes the expected payoffs. Definition 1. We say that m is an equilibrium mass if there exist a strategy σ and an endogenous arrival process χ such that E1: σ is a best response given χ, E2: m Π, m satisfies (8), and m (h, x, a) = e σ(h,x,a) P (a h, x) ˆχ (a, t (h) + 1) m (h, x), for each h, x, and a, (12) where χ and ˆχ are defined in (10) and (11), respectively. Condition E1 says that each agent acts optimally, and condition E2 ensures that the mass is feasible and consistent with the strategy. The definition of equilibrium focuses on the mass rather than the strategy because, in empirical applications, typically only the former is observed. As we explain later, the mass provides partial information about the strategy. Theorem 1. There exists an equilibrium. The proof relies on a fixed point argument. The details can be found in Appendix A. In order to characterize the equilibrium, we assume that β b P (b) <. (13) b Z The assumption is naturally satisfied in models in which time begins at t = Cost interpretation of the model. It is worth mentioning that the above model has an alternative interpretation. Instead of observing an action and a payoff shock and choosing whether to accept it, an agent chooses an effort p [0, 1] A. The intensity affects the probability of the action, which is equal to p (a) P (a h, x) ˆχ (a, t). The latter depends on the availability probability and, in case of shared actions, the acceptance decision of the other participants. There is a cost c (a, p) = log p (a) 1 that is incurred if action a occurs.

13 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 13 The cost model and the payoff shock model are behaviorally equivalent and one can go back and forth between them. All results from this paper also apply to the cost model. The two models differ in the interpretation of the behavior and one or the other may be more appropriate for a particular application. Recall that in the original model, the agent observes both the arrival of action opportunity and the action, if it occurs. In contrast, only the actions that occur are observed in the cost model. There are situations where this difference plays an role. 3. Single agent problem Next, we analyze the problem of a single agent born in period b. For this section, we treat b and the arrival process χ as fixed Choice probabilities. Each strategy σ induces the probability distribution π σ {h H : b (h) = b} over lifetime histories. The distribution is characterized by evolution equations (12), with m replaced by π σ. To simplify notation, we take π σ (h) = 0 if b (h) b. Let Π b (χ) be the set of induced probability distributions. It can be shown that Π b (χ) consists of feasible π Π such that π (b) = 1 and such that π (h, x, a) P (a h, x) ˆχ (a, t (h) + 1) π (h, x), for each h, x, and a. Conversely, each π Π b (χ) determines a strategy σ π b (h) = b, for all x, and a A, for all histories h such that σ π (h, x, a) := log P (a h, x)+log ˆχ (a, t (h) + 1) log π (a h, x) for each h, x, and a, (14) where π (a h, x) = π(h,x,a). The strategy σ and the distribution π σ are two equivalent π(h,x) descriptions of the agent s behavior Individual Utility and Entropy. For each h H, each π Π b (χ), and each arrival process χ, define functions: ρ (h, x, a) := log P (x h) + 1 {a } log P (a h, x), r (h, x, a; χ) := log ˆχ (a, t (h) + 1), H (h; π) := (1 β) log π (h), (h, x, a; π) := 1 {a = } log π ( h, x) + (1 π ( h, x)). (15)

14 14 MARCIN PĘSKI Recall the notation defined in (7). We have the following result: Theorem 2. For each π Π (χ), β b U (σ π ; b, χ) =υ π + ρ π + r (.; χ) π + [(H + ) (.; π)] π, (16) This Theorem expresses the expected payoffs as a function of the induced probability distribution π σ over lifetime histories. The payoffs are equal to a sum of (mostly) easy to interpret and compute terms. The first term is the expected and discounted systematic utility. The proof of Theorem 2 in Appendix B shows that the sum of all the other terms is equal to the expected payoff from the shocks. The second and third terms are the corrections for the exogenous and endogenous components of the availability probabilities of action opportunities. Their presence indicates that expected payoffs are increasing in the arrival rates of action opportunities. This should not be surprising: the more actions available, the greater the likelihood of accepting an action with a high payoff shock. In order to interpret the last term, notice first that H (.; π) π = (1 β) β s π (h) log π (h) s b h:t(h)=s is equal to the discounted average of the Shannon entropies of the distribution π over histories in period s b. The Shannon entropy is typically interpreted as a measure of the randomness of distribution π. We do not have an interpretation for the -term. In Section 3.4, we show that 0 in the continuous time approximation of the model. For this reason, we shall to refer to (H + ) π as an approximate entropy of π. To see the intuition behind why randomness affects expected payoff from shocks, compare two types of strategies: one that accepts as many actions as possible, and another that is selective and accepts only some actions, possibly in a history-dependent way. The first strategy leads to a more random distribution than the second one. Because all payoff shocks are positive in our model, the former leads to higher payoffs than the latter. Formula (16) is closely related to well-known results from static discrete choice models. 5 In particular, in the static logit choice model of McFadden (1978), the expected 5 There is a close relation between our model and the multinomial logit. In the latter, the logit payoff shocks are associated with all alternatives, including inaction. In the static version, let

15 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 15 value of the shock is equal to the entropy of choice probabilities. As far as we know, analogous decomposition of DDC payoffs is new. Corollary 1. For each arrival process χ, there is a unique probability density induced by the best response strategy and it is equal to π BR (b, χ) := arg max υ π + ρ π + r (.; χ) π + [(H + ) (.; π)] π. (17) π Π b (χ) Proof. The first three terms of (17) are linear in the density π. In Appendix B.2, we show that the last term is strictly concave. Thus, the objective function is strictly concave. It is easy to check that the domain of the optimization problem is convex. Thus, the solution to the maximization problem (17) is unique. The representation of optimal choice as a solution to concave optimization leads to intuitive comparative statics. For future reference, we mention that the probability of action a increases with availability probability P (a h, x) (hence with ρ (h, x, a)) and endogenous arrival probability ˆχ (a, t). The reason is simple: if an action is more likely to be available, it is also more likely to be accepted. The uniqueness of the best response strategy can be established directly. Let U (h; χ) denote the continuation value of an agent with history h (that includes the utility flow in period t (h) as well as all future continuation flows). Then, it is optimal to accept action a if the continuation payoff plus the payoff shock outweighs the continuation value without a; in other words, the best response thresholds are equal to σ BR (h, x, a; χ) = max (U (h, x, ; χ) U (h, x, a; χ), 0). (18) p (A { }) be the choice probability. It is well-known (McFadden (1978)) that the expected payoff from shocks is equal to p (a) log p (a). a A { } Here, the probability that an action is chosen is equal to p (a...) = π (a...) F (a...), and, using (4), we can write the expected payoff from shocks as p (a) (log p (a) 1 log F (a...)). a A There are two main differences: the second expression sums over proper actions only and corrects for random arrival probabilities.

16 16 MARCIN PĘSKI 3.3. Interior best responses. The next result provides conditions under which the best responses are interior. For any two (infinite) histories h, h H such that b (h) = b (h ) and t (h) = t (h ), we can say that h A h if the latter history contains more actions: if x s = x s and either a s = a s or a s = for each s. Theorem 3. Suppose that for any h, h H such that such that b (h) = b (h ),t (h) = t (h ), and h A h, υ (h) > υ (h ), P (a h, x) P (a h, x), for each x, and a A, P (x h) = P (x h ) for each x. (19a) (19b) (19c) Then, a best response strategy given any process χ is interior. The proof of Theorem 3 can be found in Appendix C. Inequality (19a) says that systematic utility decreases with each additional past action. A possible interpretation is that taking actions is costly and the cost outweighs any systematic benefit. Inequality (19b) says that an action is less likely to become available for richer histories. The interpretation is that actions crowd-out other actions Entropy in continuous time approximation. Here, we explain that the entropy correction term of (16) (as well as equation (23) below) disappears in the continuous time limit of the model (see comment after Theorem 2). Assume that time is measured in intervals of length δ > 0. We re-scale the fundamental parameters P (δ) (a h, x) = δp (a h, x), and β (δ) = β δ. We use the superscript δ to denote the values of objects defined with re-scaled parameters. Other parameters can be re-scaled as well, although this would not affect the subsequent result. Lemma 1. Suppose that δ < 1. Then, there exists a constant C < such that for 2 each χ, and each π Π b (χ) In particular, (δ) (.; π) π 0 as δ 0. (δ) (.; π) π Cβ δb δ.

17 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS Population behavior In this section, we analyze the equilibrium behavior in the population Consistent masses. For each feasible mass m Π, each shared action a, and each period t, define the mass of individuals who participate in a in period t, as m S [a, t] := m (h). h:t(h)=t,a t=a We refer to m S as the mass of shared actions. Take a strategy σ, a consistent mass m, and the associated arrival process χ. Then, m S [a, t] = m (h, x, a) = h,x:t(h)=t 1, h,x:t(h)=t 1, = a S(a)\{a} = a S(a) e σ(h,x,a) P (a h, x) ˆχ (a, t) m (h, x) χ (a, t) e σ(h,x,a) P (a h, x) m (h, x) h,x:t(h)=t 1, χ (a, t). (20) The last line does not depend on the role in the shared action. Thus, m S [a, t] = m S [a, t], for all a S (a), (21) or the masses of agents with associated roles must be equal. We refer to (21) as a balance identity. It must be satisfied by any mass that is consistent with population behavior (not necessarily in equilibrium). Let M Π be the set of feasible masses m (i.e., masses that respect the probabilities of exogenous events (6)) that, additionally, satisfy initial mass equations (8) and balance identities (21). In contrast to the single-agent case, mass m M does not uniquely identify strategy. The reason is that m contains information about the joint probability of shared actions but not about the individual probabilities with which agents accept their respective roles. Any strategy with different thresholds but the same joint acceptance probabilities is consistent with the same mass. To see it more clearly, suppose that m is consistent with an interior strategy σ (i.e., m and σ satisfy E2). Take any function φ : A Z R so that a S(a)φ (a, t) = 0 for each shared a and t and

18 18 MARCIN PĘSKI so that σ (h, x, a) := σ (h, x, a) + φ (a, t (h) + 1) 0 is a well-defined strategy. Define χ (a, t) = χ (a, t) e φ(a,t). Then, χ satisfies the definition (10) for masses m and strategy σ, condition E2 holds, and m is consistent with σ. Notice that strategies σ and σ have different payoff consequences for agents with different roles. In particular, because of (5), the expected value of the payoff shock for the agent with role a, conditional on the action occurring, is higher by φ (a, t) if using strategy σ rather than σ. The total sum of payoff changes is equal to a S(a) φ (a, t) = 0. Hence, strategy σ generate the same average welfare as σ, even if the two strategies differ in how they redistribute utility across participants Population Welfare plus Entropy. Next, we present a population version of Theorem 2: a characterization of the average welfare as a function of the history distribution. Define an index of action a in period t as For each mass m, let I (a) = k (a) 1. k (a) H S (m S ) := a,t β t I (a) m S [a, t] log m S [a, t]. (22) The above expression is equal to a weighted and discounted entropy of the mass of shared actions. The weights depend on the number of participants and they are equal to 0 for single-agent actions. Theorem 4. Assume (13). For each strategy σ, consistent mass m, and arrival process χ, we have β b U (σ; b, χ) P (b) = υ m + ρ m + [(H + ) (.; m)] m H S (m S ) + const. (23) }{{} b W (m) where the constant depends only on the parameter of the model (specifically, the birth rates P (b)). This Theorem says that population welfare can be represented as a function of the mass that is associated with the strategy. At the first sight, this observation might seem surprising given that masses do not uniquely identify strategies. However, as we explain above, strategies that induce the same mass differ with respect to the redistribution of utilities in the population, but not with respect to total welfare.

19 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 19 As in the single-agent case, Theorem 4 expresses total welfare as the sum of easy to interpret and compute terms. The first term of (23) is the average systematic utility. The remaining three terms (and the constant) are equal to the population average of payoff shocks. As in the single-agent case, the average payoff from the shocks is increasing in the availability probabilities and in the randomness of mass m. We interpret the fourth term as a measure of coordination gains. The proof of Theorem 4 shows that it is equal to the population average of endogenous arrival correction terms r (.; χ) π b (see formula (16)), where π b = m (. b (h) = b) is the normalized probability distribution over histories of agents born in period b. These correction terms increase in the endogenous arrival rate ˆχ (a, t). As we remarked in the discussion after Corollary 1, an increase in ˆχ (a, t) leads to an increase in the probability that a is accepted in period t. This raises the probability at which a is accepted, χ (a, t), which further increases ˆχ (a, t) for a S (a) \ {a}. Further, the same argument applied from the point of view of the other participants implies that the acceptance rates of the associated roles increase. In turn, this leads to a further increase in ˆχ (a, t). This positive feedback loop is a typical feature of coordination games. The coordination gains are decreasing with the randomness (measured by entropy) of the mass of actions. Intuitively, if acceptance strategies are less coordinated, the benefit from narrowly targeting particular actions is smaller, which further implies that the chosen outcomes are going to look more random. We say that m is constrained efficient if it maximizes the right-hand side of (23). Such m would be chosen by a social planner whose goal is to maximize the welfare, but who is constrained by threshold strategies Interior equilibrium. For each m, let I m S = β t I (a) m S [a, t]. (24) a A,t This is a weighted and discounted average mass of shared actions, with weights given by index function I. Theorem 5. Assume (13). If an interior mass m is an equilibrium, then it is a (constrained) critical point of the following problem: W (m) I m S st. m M, (25)

20 20 MARCIN PĘSKI where W (m) is defined in (23). Moreover, if the thesis of Theorem 3 holds (i.e., each single-agent best response is interior), then each critical point of the above problem is an interior equilibrium mass. Theorem provides a characterization of interior equilibrium masses. Each such mass is a critical point of functional (25). Moreover, if each best response is interior, then any critical point is an interior equilibrium mass. Although the characterization is not explicit, it turns out to be tractable enough for multiple purposes. There are few important consequences of the result. First, representation (25) makes it easier to find an equilibrium. Instead of looking for a potentially complicated dynamic fixed point, it is enough to find solutions to a constrained optimization problem. Computationally, this can be done either directly (for instance, using gradient methods), or by solving for the first-order conditions. An example of equilibrium is a mass that maximizes (25). In contrast to the single agent problem (17), the functional in (25) is not necessarily concave. This is due to the fact that the coordination gains term is convex in m. In particular, the maximum of (25) is possibly not the only critical point of (25). The nonuniqueness of equilibrium is inherent to search models: if more agents accept shared actions in period t, it is easier to find a good match in this period, which may make it more profitable to search for an action. Second, Theorems 4 and 5 lead to an intuitive welfare analysis. The equilibrium masses are not constrained efficient. There are two reasons for this. First, an equilibrium mass is a critical point, and not necessarily a maximizer of the functional (25). Second, even if the equilibrium mass is a maximizer, the value of the objective is smaller than welfare (23) by a term I m S that increases in the mass of shared actions. It follows that welfare would increase if agents were to accept a larger number of shared actions. The welfare loss is due to the fact that individual best responses do not internalize payoff shocks of the other agents. To see it, notice that an agent with payoff shock equal to σ (h, x, a) is indifferent between accepting or rejecting a. At the same time, conditional on the agent being pivotal, each of the other participants expects a payoff increase of 1 (see (5)). The total welfare loss if the agent rejects the action is equal to k (a) 1, or k(a) 1 per participant. k(a) Some welfare can be restored if shared actions are subsidized. Suppose that each agent receives a transfer of τ (a, t) if a is accepted (by all participants) in period t.for

21 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 21 any given m, the subsidy increases the individual systematic utility to υ (h) = υ (h) + τ ( a t(h), t (h) ), or the population welfare by τ m S. (Of course, total welfare inclusive of the social planner, is not affected by transfers.) The two theorems imply that if τ = I, then there exists an equilibrium with subsidies that is constrained efficient in the original model. Third, in Section 5.4 below, we use Theorem 5 to precisely describe what can and what cannot be identified in equilibrium. The characterization of an equilibrium as a solution to an extremal problem bears resemblance to a basic property of potential games (Monderer and Shapley (1996)). In such games, there exists a function of action profiles, called potential, with the property that each equilibrium is a local maximizer of the potential. The potential function, if it exists, has multiple applications beyond finding equilibrium (for instance, to learning and evolutionary theory, global games, etc.) Our model is not a potential game. Despite this, it is possible that functional (25) can play a role of the potential in other applications. 5. Identification This section discusses identification of model parameters from observed data: best response probabilities π in the single-agent case, and equilibrium mass m in the population case. The arrival probabilities of exogenous shocks are identified. The systematic utility and the arrival probabilities of actions are not. We show that there is a class of reduced form parameters ˆυ (υ, P ) in the single-agent case and υ (υ, P ) in the population case that are (a) sufficient to predict behavior in the model, (b) identified from data, and (c) linear in systematic utility υ and the logarithms of availability probabilities P. We use the last property to discuss counter-factual predictions Event arrival rates. The arrival probabilities of exogenous events are directly observed in data: for each history h and x, P (x h) = π (h, x) π (h) = m (h, x) m (h). (26) The arrival probabilities of actions are not directly observed because the relevant choice probabilities are affected by strategies.

22 22 MARCIN PĘSKI 5.2. Passive continuation value. A piece of notation is needed before we define the reduced-form parameters. Let σ be a strategy that never accepts any action. For each history h, let πh Π be a probability distribution over continuation histories after h that is induced by σ. The density πh can be easily calculated from the arrival probabilities of exogenous events: for each continuation h of h: t(h ) πh (h s=t(h)+1 ) = P (x s h s), if a s = for each s = t (h) + 1,..., t (h ) (27) 0, otherwise. For each f : H R, we refer to the expected discounted value f πh as the passive continuation value of f. The above remarks imply that the passive continuation values can be easily computed using observable data Single-agent identification. For each history h, let h be the history obtained from h by replacing a t(h) by. For each h, define υ (h; υ, P ) = β t(h) [(υ + ρ) πh (υ + ρ) π h ] (28) Term υ (h) is equal to the contribution of the most recent action to the passive continuation value of the systematic utility modified by availability rates. The inclusion of the discount factor β t(h) ensures that the value of the reduced form parameter is not discounted (notice that the definition of the product in (7) includes discounting). Intuitively, the definition takes into account the direct payoff effect of a as well its future consequences for the arrival of and payoffs from exogenous events. We refer to υ as a reduced-form systematic flow utility. We drop the reference to the fundamental parameters υ and P from the definition of function υ if they are clear from the context. We make the following key observation: Lemma 2. For each π Π, (υ + ρ) π = υ π + const, where the constant does not depend on π. The expected value of the modified systematic utility is equal to the expected value of the reduced-form systematic utility if the expectation is computed using any probability distribution that can feasibly arise from single-agent choices. The proof of Lemma 2 relies on restrictions imposed by the feasibility equations (6). The proof

23 UTILITY AND ENTROPY IN SOCIAL INTERACTIONS 23 does not rely on the exponential distribution of payoff shocks and likely holds more generally than the framework of this paper. Together with characterization (17), the Lemma implies that the solution to the single agent problem is equal to π BR (b, χ) = arg max π + r (.; χ) π + [(H + ) (.; π)] π. (29) π Π b (χ) υ Notice that the objective function of the maximization problem depends on parameters υ and P only through their reduced-form υ. In the interior case, the only relevant constraints of the above problem are the feasibility equations (6). It follows that an interior solution to the maximization problem depends on the original parameters υ and P only through υ and the arrival probabilities of exogenous events. In particular, nothing beyond υ and P (x h) can be identified. As a consequence, nothing beyond the reduced-form υ and the exogenous event arrival rates (26) can be identified. In particular, it is impossible to separately identify the two terms of the sum υ (h, x, a) + log P (a h, x). This is not surprising: in our model, the agent s payoffs are unchanged if the systematic utility from a history that ends with an action is reduced and the logarithm of the arrival probability of this action is increased in exactly the same way. In the single-agent case, the reduced-form parameters υ are identified. Theorem 6. Assume the single-agent version of the model, i.e., k (a) = 1 for each action a. If π is an interior solution to (29) (i.e., an interior best response density), then, for each h, υ (h) = β t(h) ((H + ) (.; π)) π h β t(h) ((H + ) (.; π)) π h (30) =: υ D (h; π). Theorem says that υ can be expressed as a function of the revealed choice probabilities and parameters that are directly observed through (26) and (27). We write υ D (h; π) for the LHS of the above equation to emphasize its dependence on data π. Theorem 6 closely corresponds to identification results from the DDC literature (see, for example, Theorem 2 in Arcidiacono, Miller, Khan, Robin, and others (2015)). In that literature, the identification of the original parameters is possible only under additional assumptions (like fixing some of the utilities at 0). Here, instead of identifying