The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Bargaining and markets: complexity and the competitive outcome

Rubinstein and Wolinsky (Rev. Econ. Stud. 57 (1990) 63-78) consider a simple decentralised market game in which agents meet randomly or voluntarily and bargain over the terms of trade. They show that any individually rational price can be sustained as a sequential equilibrium even though the model has a unique competitive outcome. Here, I consider Rubinstein and Wolinsky's model and show that if complexity costs of implementing strategies enter players’ preferences, together with the standard payoff in the game, then every equilibrium is stationary/Markov and induces the unique competitive price. Thus, I demonstrate that aversion to complexity may provide a justification for the competitive outcome.     

Generalized assignment models: with an applicationto technology transfer

Summary.  We consider a generalized assignment model where the payoffs depend on the number of matchings that take place. We formulate a simple non-cooperative game and look for subgame perfect equilibrium of this model. Existence is established for a wide class of games. We also look at a refinement criterion which, for the standard assignment model, selects the -optimal outcome as the unique equilibrium. We then apply these concepts to a model of technology transfer between domestic and foreign firms. Received: June 24, 1994; revised version October 12, 1995     

The (Price) Core of a Bertrand–Edgeworth Economy

In this paper, a solution concept from cooperative game theory is applied to a special kind of oligopolistic markets. Traditionally, oligopoly theory uses non-cooperative game theory. The cooperative solution concept, the Price Core, has been specially developed for this paper. In contrast to the Core, in the Price Core, different consumers may pay different prices for the same commodity. An example shows that the set of competitive allocations may be a proper subset of the Core and that the Core may be a proper subset of the Price Core. Received July 24, 2001; revised version received February 25, 2002 Published online: December 5, 2002     

Cooperative games in permutational structure

Summary. By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley. Received: October 31, 1995; revised version: February 5, 1997     

On the Irrelevance of Risk Attitudes in Repeated Two-Outcome Games

We study equilibrium and maximin play in supergames consisting of the sequential play of a finite collection of stage games, where each stage game has two outcomes for each player. We show that for two-player supergames in which each stage game is strictly competitive, in any Nash equilibrium of the supergame, play at each stage is a Nash equilibrium of the stage game provided preferences over certain supergame outcomes satisfy a natural monotonicity condition. In particular, equilibrium play does not depend on risk attitudes. We establish an invariance result for games with more than two players when the solution concept is subgame perfection. Journal of Economic Literature Classification Numbers: C72, C9.     

Public choice and the development of modern laboratory experimental methods in economics and political science

The paper is an account of the development of laboratory experimental methods in the early 1970s as influenced by the fields of Public Choice and Social Choice. Just a few key experiments conducted during a period when no experimental markets research was taking place, provide a bridge with the subsequent, rapid, growth of experimental economics. A new focus on public goods and externalities, as opposed to private goods traditionally used in economics experiments, required new representations of the commodity space and preference inducement methods. The importance of voting and collective decision making processes dictated the testing of equilibrium concepts from political science and cooperative game theory as opposed to the competitive equilibrium and Nash equilibria found in economics. The existence of many theories from multiple disciplines required new experimental designs and experimental tests. The Public Choice and Social Choice emphasis on comparing the performance of different types of collective decision processes induced early experiments related to institutional design and testing.     

营销渠道的冲突与协调研究——基于我国食品制造商和直接零售商合作博弈的实证分析

当渠道的非合作导致的冲突被逐渐认识到时,渠道成员就会有意识地开展合作。那么,合作是否确实减少了冲突,增进了协调呢？本文利用博弈理论提出六个假设,然后用现代计量方法进行实证检验,最后得出以下结论：渠道合作博弈优于非合作博弈,渠道成员间倾向于合作博弈,合作博弈下的营销渠道纵向成员（参与方）选择为地位相当者,渠道合作博弈中普遍存在隐性冲突,其程度与成员影响力的运用有关,合作博弈下的渠道成员普遍比较协调,营销渠道成员合作的协调及持续协调与诚信公平相关。     

Symmetrically pairwise-bargained allocations in an assignment market

Consider an assignment game with transferable utility where the optimally assigned partners engage in bargaining of the sort modelled by Nash, using as their threats the maximum they could receive in an alternative match. A symmetrically pairwise-bargained (SPB) allocation is a core allocation such that all partners are in bargained equilibrium. It is shown that an SPB allocation always exists, that the set of SPB allocations coincides with the intersection of the kernel and the core, and that there is a rebargaining process which converges to an SPB allocation if it begins at a “distinguished point” in the core.     

An admissible set occurring in various bargaining situations

Given a set of alternatives S and a binary relation M on S the admissible set of the pair (S, M) is defined to be the set of maximal elements with respect to the transitive closure of M. It is shown that existing solutions in game theory and mathematical economics are special cases of this concept (they are admissible sets of a natural S and M). These include the core of an n-person cooperative game, Nash equilibria of a noncooperative game, and the max-min solution of a two-person zero sum game. The competitive equilibrium prices of a finite exchange economy are always contained in its admissible set. Special general properties of the admissible set are discussed. These include existence, stability, and a stochastic dynamic process which leads to outcomes in the admissible set with high probability.     

Price Competition Between Teams

This study uses an experimental approach to examine whether markets are sensitive to the internal incentive structure of the competitors. Toward this goal, we modeled the competitors in a price competition duopoly game as three-player teams. Each player simultaneously declares a bid (price) and the team whose total bid was lower won the competition and was paid accordingly. The losing team was paid nothing, and in case of a tie, each team was paid half its price. This duopoly game was studied under two conditions; a cooperative treatment in which the team's profit was divided equally amongst its members and a non-cooperative one in which each individual member was paid her own bid. Whereas the Nash equilibrium is for each player in either treatment to demand the minimal price possible, we predicted that convergence to the competitive price would be much faster in the cooperative treatment than in the non-cooperative one. The experimental results firmly confirmed this prediction.     

Competitive outcomes and the inner core of NTU market games

We consider the inner core as a solution concept for cooperative games with non-transferable utility (NTU) and its relationship to payoffs of competitive equilibria of markets that are induced by NTU games. An NTU game is an NTU market game if there exists a market such that the set of utility allocations a coalition can achieve in the market coincides with the set of utility allocations the coalition can achieve in the game. In this paper, we introduce a new construction of a market based on a closed subset of the inner core which satisfies a strict positive separability. We show that the constructed market represents the NTU game and, further, has the given closed set as the set of payoff vectors of competitive equilibria. It turns out that this market is not uniquely determined, and thus, we obtain a class of markets. Our results generalize those relating to competitive outcomes of NTU market games in the literature.     

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the other?s offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).     

Profitability of corporate social responsibility in network industries

The present paper shows that, when firms compete in a non-cooperative way on the level of corporate social responsibility (CSR) in network industries, the conventional result of the prisoner’s dilemma structure of the game in standard industries—i.e. to have social concerns is the Nash equilibrium, but it is harmful for firms’ profits—vanishes and, for sufficiently intense network externalities, the equilibrium in which both firms have social concerns is more profitable than simple profit-seeking. Moreover, we show that—when firms cooperate in choosing the profit-maximising level of social concerns—a profit-maximising CSR level does exist, provided that network effects are sufficiently strong. Therefore, in network industries, firms may obtain higher profits engaging in—cooperatively as well as non-cooperatively—CSR activities, showing that firms’ social concerns may be motivated by the owners’ selfish behaviour. Finally, a counter-intuitive result as regards consumer’s surplus and social welfare is obtained: those are always higher under competitive than cooperative choice of CSR because the level of CSR activities is higher in the former case. However, given that firms gain their largest profits with the cooperative choice of CSR, a Pareto-superior outcome is not reached.     

Codetermination,Price Competition and the Network Industry

This research develops a tractable two‐stage non‐cooperative game with complete information describing the behaviour of price‐setting firms that must choose to be profit maximisers or bargainers under codetermination in a network industry with horizontal product differentiation. The existing theoretical literature has already shown that codetermination might arise as the endogenous market outcome in a strategic competitive quantity‐setting duopoly. In sharp contrast with this result, the present article shows that codetermination does never emerge as a Nash equilibrium in a price‐setting non‐network duopoly. Then, it aims at highlighting the role of network externalities in determining changes of paradigm of the game and letting codetermination become a sub‐game perfect Nash equilibrium when prices are strategic substitutes or strategic complements. This equilibrium may be Pareto efficient. Results allow distinguishing between mandatory codetermination and voluntary codetermination. The article also proposes a model of endogenous codetermination according to which every firm may choose to bargain with its own corresponding union bargaining unit only whether the firm's bargaining strength is exactly the profit‐maximising one. The equilibrium outcomes emerging in this case range from a uniform Nash equilibrium, in which both firms are codetermined, to mixed Nash equilibria, in which only one of them chooses to be codetermined. These results are ‘network depending’ and do not hold in a non‐network duopoly.     

A DYNAMIC ENTRY AND PRICE GAME WITH CAPACITY INDIVISIBILITY

Strategic market interaction is here modelled as a two‐stage game in which potential entrants choose capacities and next active firms compete in prices. Due to capital indivisibility, the capacity choice is made from a finite grid and there are economies of scale. In the simplest version of the model with a single production technique, the equilibrium turns out to depend on the ratio between the level of total output at the long‐run competitive equilibrium and the firm's minimum efficient scale: if that ratio is sufficiently large (the market is sufficiently ‘large’), then the competitive price emerges at a subgame‐perfect equilibrium of the capacity and price game; if not, then the firms randomize in prices on the equilibrium path. The role of the market size for the competitive outcome is shown to be even more important if there are several available production techniques.     

Core and equilibrium in incomplete markets

Conclusions The classical foundation of general equilibrium analysis by the cooperative concept of the core has been extended to an economy with incomplete trading possibilities. This has been accomplished by restricting the exchange possibilities of coalitions of traders in accordance with the available market structure. However, compared with the classical result, the present foundation of rational expectations equilibria may appear much weaker. The reason is that in reality consumers may sign contracts which are more complex than the existing market structure. Therefore, it may happen that prices cannot decentralize all cooperative transactions by markets, in spite of perfect competition. With an incomplete set of markets, therefore, a cooperative exchange of commodities by individual contracts may coexist with trading in non-cooperative, competitive markets.I wish to thank Martin Hellwig for helpful comments and suggestions. Of course, I remain responsible for all shortcomings of the paper.     

Markets for contracts: experiments exploring the compatibility of games and markets for games

Summmary. The research explores the relationship between games and the economic environment in which the games might be embedded. The focus is on a market institution in which agents buy and sell rights to participate in a follow-on stage of strategic interaction. The central question posed concerns how two different types of processes, the game and the market, interact. The market converges to a competitive equilibrium that is consistent with the Nash equilibrium that obtains in the game, and the convergence of the market to a competitive equilibrium lags the convergence of behaviors in the game to a Nash equilibrium.     

Nonprofits and public good provision: A contest based on compromises

Two competing nonprofits with ideologically distinct missions compete for donor funding to provide an indivisible public good in a population with heterogeneous preferences. This paper examines the extent to which (average) public values are undermined and nonprofits’ ideology compromised in a contractual game in which the right to provide the public good is the outcome of competition between nonprofits. We also scrutinize the roles of (i) cooperative versus competitive contracting, (ii) multiple public goods, (iii) enforceability of actions and (iv) observability of nonprofit costs in determining the equilibrium terms of the contract. In each case, the intensity of the ideological divide between the donor and the nonprofits jointly impact the degree to which compromises are made in terms of both the public's and nonprofit's missions, and the ability on the part of the donor to reap double (cost-saving and strategic) financial gains.