A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts

This paper explores the coordination between a supplier and a buyer within a decentralized supply chain, through the use of quantity discounts in a game theoretic model. Within this model, the players face inventory and pricing decisions. We propose both cooperative and non-cooperative approaches considering that the product traded experiences a price sensitive demand. In the first case, we study the dynamics of the game from the supplier's side as the leader in the negotiation obtaining a Stackelberg equilibrium, and then show how the payoff of this player could still improve from this point. In the second case, a cooperative model is formulated, where decisions are taken simultaneously, emulating a centralized firm, showing the benefits of the cooperation between the players. We further formulate a pricing game, where the buyer is allowed to set different prices to the final customer as a reaction to the supplier's discount decisions. For the latter we investigate the difference between feasibility of implementing a retail discount given a current coordination mechanism and without it. Finally the implications of transportation costs are analyzed in the quantity discount schedule. Our findings are illustrated with a numerical example showing the difference in the players’ payoff in each case and the optimal strategies, comparing in each case our results with existing work.     

Relative impatience determines preference between contract bargaining and repeated bargaining

In this paper we consider the effect of the `impatience ratio' I (of the worker discount factor to the firm discount factor) on the preferences of the players between two bargaining schemes in an asymmetric information wage bargaining context. The firm has private information about the worker's value and the worker makes wage demands. In the contact bargaining scheme, a wage demand which is accepted in one period is binding for all future periods (and hence the bargaining ends after acceptance of a wage demand). In the repeated bargaining scheme, the parties continue to bargain irrespective of whether the worker has been hired or not, and any accepted wage demand is only valid for the period in which it was accepted. We establish the following results under the assumption that the worker's value is uniformly distributed on an interval: When the firm is more patient than the worker (I<1) both parties prefer contract bargaining, and when the worker is more patient than the firm (I >1) both prefer repeated bargaining. For any value of I, the preferred type of bargaining gives the lower unemployment.  The work of Bae has already shown that when players are equally patient (I=1) the players are indifferent between the two schemes, regardless of the distribution of the worker's value. This paper shows that Bae's indifference result (Bae, 1991) cannot be extended to unequally patient players. Received: December 1996/Final version: October 1998     

Advertising Strategies in a Differential Game with Negative Competitor’s Interference

We consider a duopolistic industry where the current sales of each firm is proportional to its goodwill stock. The evolution of the latter depends positively on own advertising effort and negatively on competitor’s advertising. A standard assumption in the literature in differential games of advertising is that the players remain active throughout the whole (infinite) duration of the game. We relax this assumption and characterize the circumstances under which a firm finds it optimal to remain or exit the industry. Among other things, it is shown that, if both players are “strong”, then the unique Nash equilibrium is the same that one would obtain in the absence of interference from competitor’s advertising. Research supported by Ministry of University and Research of Italy, University of Padua and NSERC, Canada.     

Analysis and design of selection committees: a game theoretic secretary problem

Firms often delegate important decisions to committees which are set up specifically for that purpose; for example selection committees. We analyze the equilibrium behavior of a game in which committee members (the players) interview candidates sequentially, either hiring or going on to the next one. The players have differing evaluations of candidates (e.g. one cares about typing skills; the other about IT skills), which become their utilities if the candidate is hired. We then consider the optimal design (rules of the game) of such a committee, from the point of view of the firm. That is, which rules hire candidates which maximize the firm’s utility. Our committee game has a first round in which the members sequentially, by order of player number, say ‘yea’ or ‘nea’ to the candidate. If there are sufficient ‘yeas’ then she is tentatively hired; otherwise she is rejected. In the former case, members who said nea can veto the candidate in the second round. Thus the candidate is either hired, rejected, or vetoed. In the last case, the member casting a veto has one less to use on later candidates. We analyze equilibria where a player may say ‘yea’ to a candidate he would prefer not to hire, in order to force the other player to use up a valuable veto. We show that for the uniform candidate distribution there is a unique equilibrium and better candidates for the firm are hired when there are more vetoes. However we exhibit a candidate distribution where increasing the numbers of vetoes results in hiring worse candidates.     

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.     

On optimal intervention for customer lifetime value

The potential value of a customer to a commercial organisation can be a core ingredient in decision-making. There is a growing literature on the topic, much of it presented in rather general terms. In this paper, we examine a particular aspect, namely the expected income to the firm from a customer over his ‘lifetime’ (period of tenure with the firm). To derive quantitative results, we adopt a type of model in which there are underlying income and cost processes for a customer that evolve over time. In addition, we include a customer-specific effect that reflects the fact that some customers give rise to more income than others. These two stochastic elements are monitored by the firm for each new customer over an initial probationary period. Depending on the result of this assessment the firm might decide to offer a continuation of the account, perhaps on modified terms. The problem for the firm, which is the case addressed here, is to determine the length of the probationary period and the criterion for continuation. We set out a framework in which these two aspects may be optimised.     

Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

Motivated by the “tug‐of‐war” game studied by Peres et al. in 2009, we consider a nonlocal version of the game that goes as follows: at every step two players pick, respectively, a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount ϵ > 0 (as is done in the classical case), it is an s‐stable Levy process that chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically derive a deterministic nonlocal integrodifferential equation that we call the “infinity fractional Laplacian.” We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double‐obstacle problem, both problems having a natural interpretation as tug‐of‐war games. © 2011 Wiley Periodicals, Inc.     

Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey–Clarke–Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer’s revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players’ valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.     

Players indifferent to cooperate and characterizations of the Shapley value

In this paper we provide new axiomatizations of the Shapley value for TU-games using axioms that are based on relational aspects in the interactions among players. Some of these relational aspects, in particular the economic or social interest of each player in cooperating with each other, can be found embedded in the characteristic function. We define a particular relation among the players that it is based on mutual indifference. The first new axiom expresses that the payoffs of two players who are not indifferent to each other are affected in the same way if they become enemies and do not cooperate with each other anymore. The second new axiom expresses that the payoff of a player is not affected if players to whom it is indifferent leave the game. We show that the Shapley value is characterized by these two axioms together with the well-known efficiency axiom. Further, we show that another axiomatization of the Shapley value is obtained if we replace the second axiom and efficiency by the axiom which applies the efficiency condition to every class of indifferent players. Finally, we extend the previous results to the case of weighted Shapley values.     

A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

The setting of profit targets for target oriented divisions

Setting profit targets and striving to achieve them is fundamental to business survival and success. However, there has been little research on modeling profit-target setting. In this paper, we study analytic target setting under a common business scenario where a firm is in control of multiple divisions. Both the firm and the divisions maximize the profit probability, i.e., the probability of achieving some given profit target. The firm sets a profit target for each division which then acts as a price-setting newsvendor. We first obtain the optimal order quantity, the optimal retail price, and the maximal profit probability of a single division given its assigned target. We then derive the firm’s profit probability and focus on two specific cases to gain more managerial insights. In the first case of fair target setting, we show that when each division’s demand distribution has an increasing failure rate, if a division has a relatively high (low) production cost, its assigned profit target decreases (increases) with regard to its price elasticity. In the second case, if the firm is in control of two identical divisions, each division’s optimal profit target is just half of the firm’s profit target when the price elasticity is two or more, regardless of production cost and demand distribution.     

Some remarks on mean field games

In this article, we study three aspects of mean field games (MFG). The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of “noise” in discrete space models and the formulation of the Master Equation in this case. Finally, we show how MFG reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.     

Existence of Z-cyclic 3PDTWh(<Emphasis Type="Italic">p</Emphasis>) for Prime <Emphasis Type="Italic">p</Emphasis> ≡ 1 (mod 4)

A directed triplewhist tournament on p players over Z _p is said to have the three-person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(p). In this paper, we investigate the existence of a Z-cyclic 3PDTWh(p) for any prime p ≡ 1 (mod 4) and show that such a design exists whenever p ≡ 5, 9, 13 (mod 16) and p ≥ 29. This result is obtained by applying Weil’s theorem. In addition, we also prove that a Z-cyclic 3PDTWh(p) exists whenever p ≡ 1 (mod 16) and p < 10, 000 except possibly for p = 257, 769. Gennian Ge’s Research was supported by National Natural Science Foundation of China under Grant No. 10471127, Zhejiang Provincial Natural Science Foundation of China under Grant No. R604001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

Fictitious play in stochastic games

In this paper we examine an extension of the fictitious play process for bimatrix games to stochastic games. We show that the fictitious play process does not necessarily converge, not even in the 2 × 2 × 2 case with a unique equilibrium in stationary strategies. Here 2 × 2 × 2 stands for 2 players, 2 states, 2 actions for each player in each state.     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

Competition and Cooperation in Non-Centralized Linear Production Games

In this paper we analyze how to improve the benefits of n producers when: (1) each producer i faces a linear production problem given by max {cⁱ xⁱ:Aⁱ xⁱ≤ bⁱ,xⁱ 0}, and (2) maintaining the production capabilities of each producer is mandatory. In order to maximize the benefits, the producers decide to trade their resources while ensuring their initial individual gains. We study the games which describe this non-centralized linear production situation when players do not cooperate (section two), when players cooperate and side payments are possible (section three), and when players cooperate and side payments are not possible (section four).     

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.     

Discounted and finitely repeated minority games with public signals

We consider a repeated game where at each stage players simultaneously choose one of the two rooms. The players who choose the less crowded room are rewarded with one euro. The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced, hence the game has imperfect public monitoring. An undiscounted version of this game was considered by Renault et al. [Renault, J., Scarlatti, S., Scarsini, M., 2005. A folk theorem for minority games. Games Econom. Behav. 53 (2), 208–230], who proved a folk theorem. Here we consider a discounted version and a finitely repeated version of the game, and we strengthen our previous result by showing that the set of equilibrium payoffs Hausdorff-converges to the feasible set as either the discount factor goes to one or the number of repetition goes to infinity. We show that the set of public equilibria for this game is strictly smaller than the set of private equilibria.     

Limiting distributions of the number of pure strategy Nash equilibria in n-person games

We study the number of pure strategy Nash equilibria in a “random” n-person non-cooperative game in which all players have a countable number of strategies. We consider both the cases where all players have strictly and weakly ordinal preferences over their outcomes. For both cases, we show that the distribution of the number of pure strategy Nash equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to infinity. We also find, for each case, the distribution of the number of pure strategy Nash equilibria when the number of strategies of one player goes to infinity, while those of the other players remain finite.     

Repeated congestion games with bounded rationality

We consider a repeated congestion game with imperfect monitoring. At each stage, each player chooses to use some facilities and pays a cost that increases with the congestion. Two versions of the model are examined: a public monitoring setting where agents observe the cost of each available facility, and a private monitoring one where players observe only the cost of the facilities they use. A partial folk theorem holds: a Pareto-optimal outcome may result from selfish behavior and be sustained by a belief-free equilibrium of the repeated game. We prove this result assuming that players use strategies of bounded complexity and we estimate the strategic complexity needed to achieve efficiency. It is shown that, under some conditions on the number of players and the structure of the game, this complexity is very small even under private monitoring. The case of network routing games is examined in detail.