考虑损失厌恶行为与谈判破裂的Rubinstein谈判博弈研究

本文考虑具有损失厌恶行为与破裂风险的Rubinstein谈判博弈。首先构建子博弈完美均衡,并证明子博弈完美均衡的存在性及唯一性。然后分析子博弈完美均衡的性质,结果表明:参与人受益于对手的损失厌恶行为,而因自身具有损失厌恶行为遭受损失;谈判破裂概率对均衡结果的影响取决于贴现因子与参与人的损失厌恶系数;当谈判破裂的概率趋于零时,极限均衡结果收敛于经典的Rubinstein谈判博弈结果。最后建立了与非对称Nash谈判解的关系,其中参与人的议价能力与自身的损失厌恶水平呈负相关性,与对手的损失厌恶水平呈正相关性;参与人的议价能力依赖于谈判破裂概率与出价时间间隔的比值。     

Weighted Matching in Chess Tournaments

In many chess tournaments, e.g. when the Swiss system is used, the number of players is much larger than the number of rounds to be played. In such tournaments the pairing for a round depends on the results in earlier rounds, and the pairing process can be very complicated. In these pairing systems the main goals are to let players with equal scores play together, and that each player should alternately play white and black, with the restriction that no player may face the same opponent more than once. The paper describes how a weighted matching algorithm is used to find ‘the best pairing’ by converting the pairing rules into penalty points.     

Using dynamic programming to determine an optimal strategy in a contract bridge tournament

One form of contract bridge tournament comprises several rounds of matches in which players compete as pairs for ‘masterpoints’ awarded for each match won or drawn and for being highly placed at the end of the tournament. In the second and subsequent rounds, pairs are matched against other pairs that have been approximately equally successful so far. There has been vague discussion among bridge players of the possible advantages of ‘throwing’, that is, deliberately losing a match in round r so as to obtain a lower ranking and face an easier opponent in round r+1. This paper uses dynamic programming and data from one tournament to identify an optimal strategy (either ‘throw’ or ‘play to win’). The optimal strategy is a function of a pair's ability, its placing after each round, the relative rewards for winning a match and finishing high, the opponent's skill, and the round. For example, we find that a very weak pair should ‘throw’ and ‘play to win’ in alternate rounds.     

Time inconsistency and learning in bargaining games

This paper studies an alternating-offers bargaining game between possibly time-inconsistent players. The time inconsistency is modeled by quasi-hyperbolic discounting and the “naive backwards induction” solution concept is used in order to obtain the results. Both naive agents who remain naive and those who learn about their own preferences are considered. Offers of the players who are naive or partially naive are never accepted by any type of player in either no learning or gradual learning cases. The game between a naive or partially naive player who never learns and a time-consistent agent ends in an immediate agreement if the time-consistent agent is the proposer. A one period delay occurs if the time-consistent agent is the responder. The more naive the player is, the higher the share received. In addition, two naive agents who never learn disagree perpetually. When naive and partially naive agents play against exponential or sophisticated agents and they are able to learn their types over time, there exists a critical date before which there is no agreement. Therefore, the existence of time-inconsistent players who can learn their types as they play the game can be a new explanation for delays in bargaining. The relationship among the degree of naivete, impatience level and bargaining delay is also characterized. Specifically, for sufficiently high discount factors, agreement is always delayed. On the other hand, if the naive agent has sufficiently firm initial beliefs (slow learning or high degree of naivete), agents agree immediately.     

Finitely repeated prisoners’ dilemma experiments without a commonly known end

Using a symmetric two-player prisoners’ dilemma as base game, each player receives a signal for the number of rounds to be played with the same partner. One of these signals is the true number of rounds R while the other is R − 5. Thus both players know that the game has a finite end. They both know that the opponent knows this, but the finite end is not commonly known. As a consequence, both mutual defection and mutual cooperation until the second last round are subgame perfect equilibrium outcomes. We find experimental evidence that many players do in fact cooperate beyond their individual signal round.     

Game theory with translucent players

A traditional assumption in game theory is that players are opaque to one another—if a player changes strategies, then this change in strategies does not affect the choice of other players’ strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appropriate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i believe that everyone else would still be rational even if i were to switch strategies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax-dominated strategies, where a strategy \(\sigma \) for player i is minimax dominated by \(\sigma '\) if the worst-case payoff for i using \(\sigma '\) is better than the best possible payoff using \(\sigma \).     

Strategic risk and response time across games

Experimental data for two types of bargaining games are used to study the role of strategic risk in the decision making process that takes place when subjects play a game only once. The bargaining games are the ultimatum game (UG) and the yes-or-no game (YNG). Strategic risk in a game stems from the effect on one player’s payoff of the behavior of other players. In the UG this risk is high, while it is nearly absent in the YNG. In studying the decision making process of subjects we use the time elapsed before a choice is made (response time) as a proxy for amount of thought or introspection. We find that response times are on average larger in the UG than in the YNG, indicating a positive correlation between strategic risk and introspection. In both games the behavior of subjects with large response times is more dispersed than that of subjects with small response times. In the UG larger response time is associated with less generous and thus riskier behavior, while it is associated with more generous behavior in the YNG.     

基于CAPM框架下的创业企业风险资本融资契约设计

从组合的视角出发,考虑创业企业家的风险厌恶程度,通过等价利率将创业企业的风险资本融资契约适应到CAPM框架下,利用纳什议价解刻画双方在信息不对称下的均衡解,基于风险-收益的角度建立了最优融资契约设计模型,为创业企业融资提供了契约设计和风险资本类型选择的一个依据,并通过算例说明了模型的可行性。研究还表明,创业企业引入风险资本优于纯债务融资,而股权的分配则取决于双方的议价力。同时,风险厌恶程度越低的创业企业家越趋向于选择独立的风险投资机构;风险厌恶程度越高的创业企业家越趋向于选择公司背景的风险投资机构;而风险厌恶程度不高,考虑后期贷款的创业企业家则趋向于选择银行背景的风险投资机构。     

Product rule wins a competitive game

We consider a game that can be viewed as a random graph process. The game has two players and begins with the empty graph on vertex set . During each turn a pair of random edges is generated and one of the players chooses one of these edges to be an edge in the graph. Thus the players guide the evolution of the graph as the game is played. One player controls the even rounds with the goal of creating a so-called giant component as quickly as possible. The other player controls the odd rounds and has the goal of keeping the giant from forming for as long as possible. We show that the product rule is an asymptotically optimal strategy for both players.

     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

On expectations and the monetary stakes in ultimatum games

In an ultimatum game, player 1 makes an offer of $X from a total of $M to player 2. If player 2 accepts the offer, then player 1 is paid $(M-X) and player 2 receives $X; if player 2 rejects the offer, each gets zero. In the ultimatum game experiments reported in the literature,M is typically not more than $10 (see Forsythe, Horowitz, Savin and Sefton, 1994, hereafter FHSS; Hoffman, McCabe, Shachat and Smith, 1994, hereafter HMSS, and the literature cited therein). We report new results for 50 bargaining pairs in whichM=$100, and compare them with previous outcomes from 48 pairs withM=$10. The need for an examination of the effect of increased stakes on ultimatum bargaining is suggested by a literature survey of the effect of varying the stakes in a wide variety of decision making and market experiments over the last 33 years (Smith and Walker, 1993b). Many cases were found in which the predictions of theory were improved when the monetary rewards were increased. There were also cases in which the level of monetary rewards had no effect on the results. Consequently, it is necessary to examine the stakes question on a case by case basis. The previously reported effect of instructional changes, which define different institutional contexts, on ultimatum game outcomes, and the effect of stakes reported here, suggest a game formulation that explains changes in the behavior of both players as a result of changes in the instructional treatments. We formulate such a model and indicate how it might be further tested.     

Efficiency by trust in fairness? multiperiod ultimatum bargaining experiments with an increasing cake

Previous ultimatum bargaining experiments have shown that bargainers face the conflict whether to exploit bargaining power or to comply with basic norms of distributive justice. In multiperiod ultimatum bargaining for an increasing cake, trust in fairness can enable cooperation and thus more efficient results but is also open to opportunistic exploitation. In such a game the two players take turns in being the one who suggests an agreement and decides whether this is the final proposal, whereas his partner can only accept this proposal or reject it. While the game theoretic solution implies an immediate agreement assigning nearly all the cake to the demanding player, efficiency requires to postpone the agreement to the last possible round. Our 2 × 2-factorial design varies the number of possible bargaining periods and the cake increase, allowing us to explore several hypotheses.     

Quota games with a continuum of players

In this paper the notion ofm-quota game with a continuum of players is defined and the theory of bargaining sets is generalized to this new class of games. We discuss only the bargaining setM ₀ and our results are similar to those obtained in the finite case. Our main result is that for maximal coalition structures the stable payoff functions are exactly those in which almost every non-weak player receives no more than his quota and the weak players receive zero.     

The Kalai-Smorodinsky bargaining solution with loss aversion

We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.     

The graph grabbing game on Km,n-trees

The graph grabbing game is a two-player game on weighted connected graphs where all weights are non-negative. Two players, Alice and Bob, alternately remove a non-cut vertex from the graph (i.e., the resulting graph is still connected) and get the weight assigned to the vertex, where the starting player is Alice. Each player’s aim is to maximize his/her outcome when all vertices have been taken, and Alice wins the game if she gathered at least half of the total weight. Seacrest and Seacrest (2017) proved that Alice has a winning strategy for every weighted tree with even order, and conjectured that the same statement holds for every weighted connected bipartite graph with even order. In this paper, we prove that Alice wins the game on a type of a connected bipartite graph with even order called a $K_{m,n}$-tree.     

Location game on disjoint line segments

Two players are endowed with resources for setting up N locations on K line segments of identical length, with N > K ≥ 1. The players alternately choose these locations (possibly in batches of more than one in each round) in order to secure the area closer to their locations than that of their rival’s. The player with the highest secured area wins the game and otherwise the game ends in a tie. Earlier research has shown that, if an analogous game is played on disjoint circles, the second mover advantage is in place only if K = 1, while for K > 1 both players have a tying strategy. It was also shown that these results hold for line segments of identical length when rules of the game additionally require players to take exactly one location in the first round. In this paper we show that the second mover advantage is still in place for K ≥ 1 and 2K − 1 ≤ N, even if the additional restriction is dropped, while K ≤ N < 2K − 1 results in the first mover advantage. Our results allow us to draw conclusions about a natural variant of the game, where the resource mobility constraint is more stringent so that in each round each player chooses a single location and we show that the second mover advantage re-appears for K ≤ N < 2K − 1 if K is an even number. In all the cases the losing player has a strategy guaranteeing him arbitrarily small loss.     

An <Emphasis Type="Italic">m</Emphasis><Superscript>th</Superscript> Generation Model for Weak Ranking of Players in a Tournament

The problem of ranking players in a tournament has been studied by a number of authors. The methods developed for ranking players fall under two general headings—direct methods and rating methods. The present paper extends the tournament ranking problem in two directions. First, the usual definition of a tournament is broadened to include ties or draws. Thus, our model determines the best weak ranking of the players. Second, the ranking method presented takes account of player strength in that wins over strong players are valued higher than wins over weak players. To account for player strength, we evaluate both direct or first-order wins of players over opponents (i defeats j) and indirect or higher-order wins (i defeats k, who defeats j). A model which derives a composite score for each player, combining both direct and indirect wins, is used to obtain an overall ranking of the competitors.     

A generalization of the Egalitarian and the Kalai–Smorodinsky bargaining solutions

We characterize the class of weakly efficient n-person bargaining solutions that solely depend on the ratios of the players’ ideal payoffs. In the case of at least three players the ratio between the solution payoffs of any two players is a power of the ratio between their ideal payoffs. As special cases this class contains the Egalitarian and the Kalai–Smorodinsky bargaining solutions, which can be pinned down by imposing additional axioms.     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

Vertical cooperative advertising and pricing decisions in a manufacturer–retailer supply chain: A game-theoretic approach

Manufacturers can increase the advertising expenditures of their retailers by bearing a fraction of the occurring costs within the framework of a vertical cooperative advertising program. We expand the existing research which deals with advertising and pricing decisions in a manufacturer–retailer supply chain contemporaneously. By means of game theory, four different relationships between the channel members are considered: Firstly, three non-cooperative games with either symmetrical distribution of power or asymmetrical distribution with one player being the leader in each case, and one cooperative game where both players tend to maximize the total profit. The latter is complemented by a bargaining model, which proposes a fair split of profit on the basis of the players’ risk attitude and bargaining power. Our main findings are as follows: (a) In contrast to previous analyses, we do not limit the ratio between manufacturer’s and retailer’s margin, which provides more general insights into the effects of the underlying distribution of power within the channel. (b) The highest total profit is gained when both players cooperate. This behavior puts also the customers in a better position, as it produces the lowest retail price as well as the highest advertising expenditures compared to the other configurations.