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This paper introduces a family of domains of bargaining problems allowing for non-convexity. For each domain in this family, single-valued bargaining solutions satisfying the Nash axioms are explicitly characterized as solutions of the iterated maximization of Nash products weighted by the row vectors of the associated bargaining weight matrices. This paper also introduces a simple procedure to standardize bargaining weight matrices for each solution into an equivalent triangular bargaining weight matrix, which is simplified and easy to use for applications. Furthermore, the standardized bargaining weight matrix can be recovered from bargaining solutions of simple problems. This recovering result provides an empirical framework for determining the bargaining weights.

Competitive implications of cross-holdings have been extensively analyzed in the literature. Incentives for engaging cross-holdings and welfare effects were however rarely studied. Although a similar logic as with the merger paradox holds for Cournot oligopolies with homogeneous products and symmetric technologies, we show that there are profit incentives for firms to engage cross-holdings with asymmetric technologies. Furthermore, we show that social welfare could be enhanced with cross-holdings even though the market becomes more concentrated. We also discuss the robustness of both the submodularity of the Cournot model with respect to the presence of cross-holdings and our results with respect to product differentiation.

Nash solutions for two-player bargaining problems with non-convexity are shown to be dictatorial selections of Nash product maximizers in recent literature. In this paper we show that these solutions are implementable as unique subgame perfect equilibrium payoff allocations of a sequential game.

A sequential variation of the Arrow–Debreu abstract economy is developed to closely capture the timing of moves of the Walrasian general equilibrium model. Instead of inducing a pseudo- game, the extensive form game of our sequential variation is well defined. It is shown that when information is symmetric, Walrasian equilibrium allocations are equivalent to subgame-perfect equilibrium allocations. When information is asymmetric, rational expectations equilibrium allocations are shown to be equivalent to perfect Bayesian equilibrium allocations. These results are useful for understanding and characterizing Walrasian and rational expectations equilibrium allocations.

We consider a network game based on matching pennies with two types of agents, conformists and rebels. Conformists prefer to match the action taken by the majority of her neighbors while rebels like to match the minority. We investigate the simultaneous best response dynamic focusing on the lengths of limit cycles (LLC for short). We show that $$\hbox {LLC}=1$$ or 2 when all agents are of the same type, and $$\hbox {LLC}=4$$ when there is no conformist-rebel edge and no two even-degreed agents (if any) are neighboring each other. Moreover, $$\hbox {LLC}=1$$ for almost all type configurations when the network is a line or a ring, which implies that a pure strategy Nash equilibrium is reached from any initial action profile. However, $$\hbox {LLC}=4$$ for about one half of the type configurations with star networks.

We analyze a model of evolution of fashion based on a social network game, to be referred to as a fashion game. Each agent in this model is either a conformist or a rebel. A conformist prefers to take the action that is most common among her neighboring agents, whereas a rebel prefers the opposite. When there is only one type of agents, the fashion game possesses an exact potential function and, thus, fashion cycles are unlikely to emerge. This confirms the insightful observation of Simmel (1904) that the heterogeneity of consumers is essential to the emergence of fashion cycles. However, the population composition is not so important. What matters is a more subtle structure factor called the homophily index. Our main finding is that a higher homophily index generally inhibits the emergence of fashion cycles. This is established through the potential analysis, the partial potential analysis, and the stability analysis of a system of ordinary differential equations that is transformed from the stochastic best response dynamic of the fashion game by applying mean-field approximations and the diffusion approximation. In particular, the partial potential analysis is introduced in this paper as an extension of the seminal potential analysis.

Games with strategic substitutes (GSS) are generally less tractable than games with strategic complements (GSC). This paper revisits the GSC versus GSS comparison by establishing a novel connection between them. We show through a network perspective that, when the strategy set of each player is the product of some linearly ordered sets that are order isomorphic to subsets of the real space, every GSC can be embedded into a GSS, such that the set of pure strategy Nash equilibria of the former is a projection of that of the latter. In comparison, no GSS with multiple pure strategy Nash equilibria can be embedded into any GSC. In this sense, the class of GSS is broader than the class of GSC.

Shapley (1955) introduced the model of an abstract market game as a generalization of the assignment game model, among several other models. He conjectured that abstract market games possess non-empty cores. We analyze properties of abstract market games and provide a proof of this conjecture for cases with four or fewer players. We show by example that, in general, the structure of an abstract market game is not strong enough to guarantee the nonemptiness of the core. We establish supplemental conditions for the conjecture to hold. Our supplemental conditions are satisfied by the assignment games and abstract market games with one side consisting of a single player as with package auction games in Ausubel and Milgrom (2002) .

We show that a high-cost firm may be induced to compete less aggressively when its ownership is increasingly held by a low-cost firm. With asymmetric information on technologies, cross-holdings may make it desirable for both the high- and low-cost firms to reveal private information.

We establish axioms under which a bargaining solution can be found by the maximization of the CES function and is unique up to specifications of the distribution and elasticity parameters. This solution is referred to as the CES solution which includes the Nash and egalitarian solutions as special cases. Next, we consider a normalization of the CES function and establish axioms, under which a bargaining solution can be found by the maximization of the normalized CES function and is unique up to specifications of the distribution and substitution parameters. We refer to this solution as the normalized CES solution which includes the Nash and Kalai-Smorodinsky solutions as special cases. Our paper contributes to bargaining theory by establishing unified characterizations of existing as well as a great variety of new bargaining solutions.

An adequate description of economic dynamics requires the introduction of a monetary system including default penalties and expectations in a society whose economy utilizes money and credit. This essay notes and discusses several of the factors involved in the use of money and credit in a process oriented economy. It links these observations with the general equilibrium treatment of the same underlying economy and formulates a government guidance game where the government sets several key parameters in a monetary economy sufficient to select a unique equilibrium. Low information and error correction are noted. The links to the first and second welfare theorems of GE are also considered as is the setting of the price level.

We consider bargaining problems with compact star-shaped choice sets arising from a class of economic bargaining environments. Convex or comprehensive (relative to the disagreement point) problems are star-shaped but not conversely. We characterize single-valued solutions satisfying the Nash axioms on the class of compact star-shaped bargaining problems. For the case with two players, we show that there are exactly two solutions with each being a dictatorial (in favor of one player) selection of Nash product maximizers. We provide an extensive form game to implement Nash bargaining solutions. We extend our analysis and results to allow for alternative domains, asymmetries, and more than two players. For the n-player case, Nash solutions are shown to be determined by n-round iterative maximizations of Nash products.

We propose a framework of consistent finite-order priors to facilitate the incorporation of higher-order uncertainties into Bayesian game analysis, without invoking the concept of a universal type space. Several recent models, which give rise to stunning results with higher-order uncertainties, turn out to operate with certain consistent order-2 priors. We introduce canonical representations of consistent finite-order priors, which we apply to establish a criterion for determining the orders of strategically relevant beliefs for abstract Harsanyi type spaces. We derive finite-order projections of type spaces and discuss convergence of BNEs based on them as the projection order increases. Finally, we introduce finite-order total variation distances between priors, which are suitable for analyzing the issues on equilibrium continuity and robustness. We revisit recent advancements of Bayesian game theory and develop new insights based on our framework.

We analyze welfare effect of information acquisition for a model of competitive financial markets with diverse information and rational expectations. We show that in the fully revealing rational expectations equilibrium, each agent’s gain from trade in ex ante utility decreases as more agents become informed. An implication of the result is that market efficiency and ex ante Pareto optimality are not compatible in competitive financial markets with diverse information and rational expectations. Our result can be viewed as complementary to the Grossman paradox, which shows that market efficiency and individuals’ incentives to acquire information are not compatible. This paper is the first step in a projected exploration of welfare effect of information acquisition in models with diverse information.

This paper analyzes the role of price as a signal of the quality of a monopoly firm's new product. The quality of the goods is drawn from a continuum and is unknown to consumers. We establish a unique separating equilibrium using equilibrium characterization results for signaling games. The equilibrium price monotonically increases with quality levels and exceeds the complete-information monopoly price for all quality levels but the lowest one. However, the upward distortion decreases as the proportion of pre-informed consumers increases. These results extend both the signaling role of price and characteristics of the separating equilibrium as established in Bagwell and Riordan (1991).

The enlargement of the general-equilibrium structure to allow for default subject to penalties results in a construction of a simple mechanism for selecting a unique competitive equilibrium. We consider economies for which a common credit money can be applied to uniquely select each of the competitive equilibria with suitable default penalties. We identify two classes of such economies.

This paper analyzes the role of seller-induced shortage as a signal of quality. Unlike dissipative advertising, the cost of inducing shortage is different for different quality types. It is shown that under certain conditions, a high-quality monopoly firm that signals quality by inducing shortage makes more profit than using price alone or combined with dissipative advertising. This is because the forgone profit from the lost sales is always lower for the high-quality firm than for the low-quality firm. The result explains why high-quality firms may prefer to initially limit supply with a price weakly lower than that in the complete-information case.

This paper considers an approach for inducing cooperation in prisoner¡¯s dilemma. The approach is based on players individually committing to pay self-stipulated penalties for defection. We provide a complete characterization of self-stipulated penalties that are necessary and sufficient to induce the players to cooperate in subgame-perfect equilibrium. An alternative interpretation of the conditions using contract remedies is provided.

We introduce endogenous fees for participating in second-price auction which we use for a two-stage mechanism to solve King Solomon’s dilemma. They are positive for all agents. They are nonetheless shown to maintain the agents’ incentives for truthful bidding and guarantee participation by the highest-value agent. This feature of the endogenous fees is powerful enough for the efficient outcome to uniquely result from one round elimination of weakly dominated strategies, followed by at most four rounds of iterative elimination of strictly dominated stage-strategies. We provide an extension to cases with n agents and k identical prizes.

© 2015. A linkage to reconcile measurable utility derived from intensity comparisons or from probability mixtures is provided in this note. This brief note is in honor of Lloyd Shapley whose relatively unknown seminal paper on measurable utility from axioms involving the fineness of perception offered a different view on utility measurement.

© 2015, Springer-Verlag Berlin Heidelberg. We introduce log-convexity for bargaining problems. With the requirement of some basic regularity conditions, log-convexity is shown to be necessary and sufficient for Nash’s axioms to determine a unique single-valued bargaining solution up to choices of bargaining powers. Specifically, we show that the single-valued (asymmetric) Nash solution is the unique solution under Nash’s axioms without that of symmetry on the class of regular and log-convex bargaining problems, but this is not true on any larger class. We apply our results to bargaining problems arising from duopoly and the theory of the firm. These problems turn out to be log-convex but not convex under familiar conditions. We compare the Nash solution for log-convex bargaining problems with some of its extensions in the literature.

Games with strategic substitutes (GSS) are generally more intractable than games with strategic complements (GSC). This well-known fact is revisited from two new perspectives, a network one and a computational one. From the network perspective, it is shown that every GSC, under certain mild conditions, can be embedded into a larger GSS, such that the set of the pure strategy Nash equilibria of the former is the projection of that of the latter. From the computational perspective, it is shown that the two focal equilibria of a GSC with linear best responses can be computed in polynomial time, while computing a pure strategy Nash equilibrium of a GSS with linear best responses is PPAD-hard. These results indicate that strategic substitutability is more fundamental than strategic complementarity, in the sense that the class of GSS is much broader than the class of GSC. Combined with the previous results in the literature, our paper provides a more complete picture about the relationship between GSC and GSS.

A model of coalition government formation is presented in which inefficient, non-minimal winning coalitions may form in Nash equilibrium. Predictions for five games are presented and tested experimentally. The experimental data support potential maximization as a refinement of Nash equilibrium. In particular, the data support the prediction that non-minimal winning coalitions occur when the distance between policy positions of the parties is small relative to the value of forming the government. These conditions hold in games 1, 3, 4 and 5, where subjects played their unique potential-maximizing strategies 91, 52, 82 and 84 percent of the time, respectively. In the remaining game (Game 2) experimental data support the prediction of a minimal winning coalition. Players A and B played their unique potential-maximizing strategies 84 and 86 percent of the time, respectively, and the predicted minimal-winning government formed 92 percent of the time (all strategy choices for player C conform with potential maximization in Game 2). In Games 1, 2, 4 and 5 over 98 percent of the observed Nash equilibrium outcomes were those predicted by potential maximization. Other solution concepts including iterated elimination of weakly dominated strategies and strong/coalition-proof Nash equilibrium are also tested.

We specify an adjustment process that converges to the set of potential-maximizing strategy profiles for 3-player cooperation-formation games or n-player cooperation-formation games based on a superadditive characteristic function. Our analysis provides a justification for potential maximization as a refinement of Nash equilibrium in these settings.

This paper looks at cooperation structures that result from a strategic game where players make simultaneous proposals for cooperation. We identify cooperation structures that maximize the potential of the game, and show how the outcome of potential maximization depends on the players' Shapley values. We do not assume superadditivity and hence, potential-maximizing strategy profiles do not always involve full cooperation. In cases where full cooperation does result from potential maximization it can be inefficient. An example provides intuition.

We analyze sunspot-equilibrium prices in nonconvex economies with perfect markets and a continuous sunspot variable. Our primary result is that every sunspot equilibrium allocation can be supported by prices that, when adjusted for probabilities, are constant across states. This result extends to the case of a finite number of equally probable states under weak nonsatiation conditions, but does not extend to general discrete state spaces. We use our primary result to establish the equivalence of the set of sunspot equilibrium allocations based on a continuous sunspot variable and the set of lottery equilibrium allocations

We apply an intermediation game of Townsend (1983) to analyze trade in an exchange economy through endogenous intermediaries. In this game, each trader has the opportunity to become an intermediary by oering to buy or sell unlimited quantities of the commodities at a certain price vector and for a certain group of customers subject to feasibility constraint. An intermediary will not be active unless some of its customers subsequently choose to trade with it. We introduce an "intermediation core" and show that the subgame-perfect equilibrium allocations of the intermediation game are contained in the intermediation core, similar to the inclusion of competitive equilibrium allocations in the core usually studied. We also identify, in terms of the supporting intermediary structures, intermediation core allocations which are also subgame-perfect equilibrium allocations of the intermediation game. These results provide both a characterization and welfare properties of subgame-perfect equilibrium allocations of the intermediation game.

We study economic natural selection in classical oligopoly settings. When underlying pure strategies consist of a finite number of prices, convex monotonic dynamics always converge under a weak condition to the smallest price in the support of the initial state that exceeds marginal cost. When underlying pure strategies consist of a finite number of quantities, monotonic dynamics always converge under a specific condition to a quantity equal or similar to classical Cournot equilibrium.

Given a game, the set of joint lotteries over partitions of the agents of any subgame induces a subset of the vectors of balancing weights for the subgame. Games whose subgames are all balanced with respect to these vectors of balancing weights are called totally L-balanced games. We show that such games are precisely the ones that can be generated from direct lottery markets. Total L-balancedness is equivalent to superadditivity. Thus, many interesting games that are not totally balanced, but are superadditive, can be generated from direct lottery markets. We also show that the core of a game coincides with the set of lottery equilibrium utility vectors of its direct lottery market.Journal of Economic LiteratureClassification Numbers: C71, D51.

We consider a network game based on matching pennies with two types of agents, conformists and rebels. Conformists prefer to match the action taken by the majority of her neighbors while rebels like to match the minority. We investigate the simultaneous best response dynamic focusing on the lengths of limit cycles (LLC for short). Our results imply that the network structure generally plays a very crucial role in determining LLC. When the network is a line or a ring, for almost all type configurations, LLC=1, meaning that a pure strategy Nash equilibrium is reached, regardless of the initial action profiles. However, when the network is a star, LLC=4 for approximately half of the type configurations. The ordinal potential property and the sequential best response dynamic are also studied.

I consider a cooperation-formation game in which players choose independently with whom they wish to cooperate in a given coalitional game, and players' payoffs follow a solution imposed on the coalitional game. I study the resulting equilibrium cooperation structure and how cooperation evolves under best-response and fictitious-play learning processes.Journal of Economic LiteratureClassification Numbers: C71, C72.

Lotteries are introduced in the exchange model of Shapley and Scarf ( J. Math. Econ. 1 (1974), 23-37). In competition, traders buy and sell probabilities on houses. When considering the core, feasible allocations of the economy and blocking allocations of coalitions may involve lotteries. Among other results, we show that the set of lottery equilibrium allocations is non-empty and does not contain all competitive equilibrium allocations, the core of the Shapley/Scarf model is a strict subset of the lottery core, and lottery equilibrium allocations are contained in the lottery core. Journal of Economic Literature Classification Numbers: C62, C71.

Sufficient conditions for a general n-person game to have a nonempty inner core are established. They relate to the balancedness condition and characterize a class of NTU games that is larger than the class of compactly generated and balanced with slack games. The inner core of a game coincides with the inner core of its cover. Moreover, it shrinks under convex utility transformations of the game. Journal of Economic Literature Classification Number: C71.

We extend the notion of the inner core of a finite economy to a large economy. We prove that competitive allocations and the core coincide with the inner core.

It is shown that for every NTUmarket game, there is amarket thatrepresents the game whosecompetitive payoff vectors completely fill up theinner core of the game. It is also shown that for every NTU market game and for any point in its inner core, there is a market that represents the game and further has the given inner core point as itsunique competitive payoff vector. These results prove a conjecture of Shapley and Shubik.Journal of Economic Literature Classification Numbers: C71, D51.

The inner core, motivated by the study of competitive outcomes in the cores of market games, is shown to be contained in the strictly inhibitive set, and coincides with the strictly inhibitive set for a large class of games. Journal of Economic Literature Classification Numbers: C71, C78.