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In online advertising markets, advertisers often purchase ad placements through bidding in repeated auctions based on realized viewer information. We study how budget-constrained advertisers may compete in such sequential auctions in the presence of uncertainty about future bidding opportunities and competition. We formulate this problem as a sequential game of incomplete information, in which bidders know neither their own valuation distribution nor the budgets and valuation distributions of their competitors. We introduce a family of practical bidding strategies we refer to as adaptive pacing strategies, in which advertisers adjust their bids according to the sample path of expenditures they exhibit, and analyze the performance of these strategies in different competitive settings. We establish the asymptotic optimality of these strategies when competitors’ bids are independent and identically distributed over auctions, but also when competing bids are arbitrary. When all the bidders adopt these strategies, we establish the convergence of the induced dynamics and characterize a regime (well motivated in the context of online advertising markets) under which these strategies constitute an approximate Nash equilibrium in dynamic strategies: the benefit from unilaterally deviating to other strategies, including ones with access to complete information, becomes negligible as the number of auctions and competitors grows large. This establishes a connection between regret minimization and market stability, by which advertisers can essentially follow approximate equilibrium bidding strategies that also ensure the best performance that can be guaranteed off equilibrium. This paper was accepted by Noah Gans, stochastic models and simulation.

We study dynamic unstructured bargaining with deadlines and one-sided private information about the amount available to share (the “pie size”). Using mechanism design theory, we show that given the players’ incentives, the equilibrium incidence of bargaining failures (“strikes”) should increase with the pie size, and we derive a condition under which strikes are efficient. In our setting, no equilibrium satisfies both equality and efficiency in all pie sizes. We derive two equilibria that resolve the trade-off between equality and efficiency by favoring either equality or efficiency. Using a novel experimental paradigm, we confirm that strike incidence is decreasing in the pie size. Subjects reach equal splits in small pie games (in which strikes are efficient), while most payoffs are close to either the efficient or the equal equilibrium prediction, when the pie is large. We employ a machine learning approach to show that bargaining process features recorded early in the game improve out-of-sample prediction of disagreements at the deadline. The process feature predictions are as accurate as predictions from pie sizes only, and adding process and pie data together improves predictions even more. Data are available at https://doi.org/10.1287/mnsc.2017.2965 . This paper was accepted by Uri Gneezy, behavioral economics.

It is not fully understood why we cooperate with strangers on a daily basis. In an increasingly global world, where interaction networks and relationships between individuals are becoming more complex, different hypotheses have been put forward to explain the foundations of human cooperation on a large scale and to account for the true motivations that are behind this phenomenon. In this context, population structure has been suggested to foster cooperation in social dilemmas, but theoretical studies of this mechanism have yielded contradictory results so far; additionally, the issue lacks a proper experimental test in large systems. We have performed the largest experiments to date with humans playing a spatial Prisoner’s Dilemma on a lattice and a scale-free network (1,229 subjects). We observed that the level of cooperation reached in both networks is the same, comparable with the level of cooperation of smaller networks or unstructured populations. We have also found that subjects respond to the cooperation that they observe in a reciprocal manner, being more likely to cooperate if, in the previous round, many of their neighbors and themselves did so, which implies that humans do not consider neighbors’ payoffs when making their decisions in this dilemma but only their actions. Our results, which are in agreement with recent theoretical predictions based on this behavioral rule, suggest that population structure has little relevance as a cooperation promoter or inhibitor among humans.

We analyze participation in international environmental agreements in a dynamic game in which countries pollute and invest in green technologies. If complete contracts are feasible, participants eliminate the holdup problem associated with their investments; however, most countries prefer to free ride rather than participate. If investments are non-contractible, countries face aholdup problem every time they negotiate; but the free-rider problem can be mitigated and significant participation is feasible. Participation becomes attractive because only large coalitions commit to long-term agreements that circumvent the holdup problem. Under well-specified conditions even the first-best outcome is possible when the contract is incomplete.

We consider a dynamic game with asymmetric information where each player privately observes a noisy version of a (hidden) state of the world V, resulting in dependent private observations. We study the structured perfect Bayesian equilibria (PBEs) that use private beliefs in their strategies as sufficient statistics for summarizing their observation history. The main difficulty in finding the appropriate sufficient statistic (state) for the structured strategies arises from the fact that players need to construct (private) beliefs on other players' private beliefs on V, which, in turn, would imply that one needs to construct an infinite hierarchy of beliefs, thus rendering the problem unsolvable. We show that this is not the case: each player’s belief on other players' beliefs on V can be characterized by her own belief on V and some appropriately defined public belief. We then specialize this setting to the case of a Linear Quadratic Gaussian (LQG) non-zero-sum game, and we characterize structured PBEs with linear strategies that can be found through a backward/forward algorithm akin to dynamic programming for the standard LQG control problem. Unlike the standard LQG problem, however, some of the required quantities for the Kalman filter are observation-dependent and, thus, cannot be evaluated offline through a forward recursion.

We introduce a dynamic game of outbidding where two groups use violence to compete in a tug-of-war fashion for evolving public support. We fit the model to the canonical outbidding rivalry between Hamas and Fatah using newly collected data on Palestinian public support for these groups. Competition has heterogeneous effects, and we demonstrate that intergroup competition can discourage violence. Competition from Hamas leads Fatah to use more terrorism than it would in a world where Hamas abstains from terrorism, but competition from Fatah can lead Hamas to attack less than it otherwise would. Likewise, making Hamas more capable or interested in competing increases overall violence, but making Fatah more capable or interested discourages violence on both sides. These discouragement effects of competition on violence emerge through an asymmetric contest, in which we find that Fatah uses terrorism more effectively to boost its support, although Hamas has lower attack costs. Expanding on these results, we demonstrate that outbidding theory is consistent with a positive, negative, or null relationship between measures of violence and incentives to compete.

This article deals with the identification and estimation of dynamic games when players’ beliefs about other players’ actions are biased, that is, beliefs do not represent the probability distribution of the actual behaviour of other players conditional on the information available. First, we show that an exclusion restriction, typically used to identify empirical games, provides testable non-parametric restrictions of the null hypothesis of equilibrium beliefs in dynamic games with either finite or infinite horizon. We use this result to construct a simple Likelihood Ratio test of equilibrium beliefs. Second, we prove that this exclusion restriction, together with consistent estimates of beliefs at two points in the support of the variable involved in the exclusion restriction, is sufficient for non-parametric point-identification of players’ belief functions as well as useful functions of payoffs. Third, we propose a simple two-step estimation method. We illustrate our model and methods using both Monte Carlo experiments and an empirical application of a dynamic game of store location by retail chains. The key conditions for the identification of beliefs and payoffs in our application are the following: (1) the previous year’s network of stores of the competitor does not have a direct effect on the profit of a firm, but the firm’s own network of stores at previous year does affect its profit because the existence of sunk entry costs and economies of density in these costs; and (2) firms’ beliefs are unbiased in those markets that are close, in a geographic sense, to the opponent’s network of stores, though beliefs are unrestricted, and potentially biased, for unexplored markets which are farther away from the competitors’ network. Our estimates show significant evidence of biased beliefs. Furthermore, imposing the restriction of unbiased beliefs generates a substantial attenuation bias in the estimate of competition effects.

We study the Markov perfect equilibrium in a dynamic game where agents have non-constant time preference, decentralized households determine aggregate savings, and a planner chooses climate policy. The article is the first to solve this problem with general discounting and general functional forms. With time-inconsistent preferences, a commitment device that allows a planner to choose climate policy for multiple periods is potentially very valuable. Nevertheless, our quantitative results show that while a permanent commitment device would be very valuable, the ability to commit policy for “only” 100 years adds less than 2% to the value of climate policy without commitment.We solve a log-linear version of the model analytically, generating a formula for the optimal carbon tax that includes the formula in Golosov et al. (2014, Econometrica, 82, 41–88) as a special case. More importantly, we develop new algorithms to solve the general game numerically. Convex damages lead to strategic interactions across generations of planners that lower the optimal carbon tax by 45% relative to the scenario without strategic interactions.

It is common for researchers studying infinite horizon dynamic games in a lab experiment to pay participants in a variety of ways, including but not limited to outcomes in all rounds or for a randomly chosen round. We argue that these payment schemes typically induce different preferences over outcomes than those of the target game, which in turn would typically implement different outcomes for a large class of solution concepts (e.g., subgame perfect equilibria, Markov equilibria, renegotiation-proof equilibria, rationalizability, and non-equilibrium behavior). For instance, paying subjects for all rounds generates strong incentives to behave differently in early periods as these returns are locked in. Relatedly, a compensation scheme that pays subjects for a randomly chosen round induces a time-dependent discounting function. Future periods are discounted more heavily than the discount rate in a way that can change the theoretical predictions both quantitatively and qualitatively. We rigorously characterize the mechanics of the problems induced by these payment methods, developing measures to describe the extent and shape of the distortions. Finally, we prove a uniqueness result: paying participants for the last (randomly occurring) round, is the unique scheme that robustly implements the predicted outcomes for any infinite horizon dynamic game with time separable utility, exponential discounting, and a payoff-invariant solution concept.

The analysis of dynamic games hinges on assumptions about players’ actions and beliefs at information sets that are not expected to be reached during game play. Under the standard notion of sequential rationality, these assumptions cannot be tested on the basis of observed, on-path behavior. This paper introduces a novel optimality criterion, structural rationality, which addresses this concern. In any dynamic game, structural rationality implies weak sequential rationality (Reny (1992)). If players are structurally rational, assumptions about on-path and off-path beliefs concerning off-path actions can be tested via suitable “side bets.” Structural rationality also provides a theoretical rationale for the use of a novel version of the strategy method (Selten (1967)) in experiments.