Establishing Nash equilibrium of the manufacturer-supplier game in supply chain management

We study a game model of multi-leader and one-follower in supply chain optimization where n suppliers compete to provide a single product for a manufacturer. We regard the selling price of each supplier as a pre-determined parameter and consider the case that suppliers compete on the basis of delivery frequency to the manufacturer. Each supplier’s profit depends not only on its own delivery frequency, but also on other suppliers’ frequencies through their impact on manufacturer’s purchase allocation to the suppliers. We first solve the follower’s (manufacturer’s) purchase allocation problem by deducing an explicit formula of its solution. We then formulate the n leaders’ (suppliers’) game as a generalized Nash game with shared constraints, which is theoretically difficult, but in our case could be solved numerically by converting to a regular variational inequality problem. For the special case that the selling prices of all suppliers are identical, we provide a sufficient and necessary condition for the existence and uniqueness of the Nash equilibrium. An explicit formula of the Nash equilibrium is obtained and its local uniqueness property is proved

Selection methods for subgame perfect Nash equilibrium in a continuous setting

This thesis is focused on the issue of selection of Subgame Perfect Nash Equilibrium (SPNE) in the class of one-leader N-follower two-stage games where the players have a continuum of actions. We are mainly interested in selection methods satisfying the following significant features in the theory of equilibrium selection for such a class of games: obtaining an equilibrium selection by means of a constructive (in the sense of algorithmic) and motivated procedure, overcoming the difficulties due to the possible non-single-valuedness of the followers' best reply correspondence, providing motivations that would induce players to choose the actions leading to the designed selection, and revealing the leader to know the followers' best reply correspondence. Firstly, we analyze the case where the followers' best reply correspondence is assumed to be single-valued: in this case we show that finding SPNEs is equivalent to find the Stackelberg solutions of the Stackelberg problem associated to the game. Moreover, as regards to the related arising issue of the sufficient conditions ensuring the uniqueness of the followers' best reaction, we prove an existence and uniqueness result for Nash equilibria in two-player normal-form games where the action sets are Hilbert spaces and which allows the two compositions of the best reply functions to be not a contraction mapping. Furthermore, by applying such a result to the class of weighted potential games, we show the (lack of) connections between the Nash equilibria and the maximizers of the potential function. Then, in the case where the followers' best reply correspondence is not assumed to be single-valued, we examine preliminarily the SPNE selections deriving by exploiting the solutions of broadly studied problems in Optimization Theory (like the strong Stackelberg, the weak Stackelberg and the intermediate Stackelberg problems associated to the game). Since such selection methods, although behaviourally motivated, do not fit all the desirable features mentioned before, we focus on designing constructive methods to select an SPNE based on the Tikhonov regularization and on the proximal point methods (linked to the Moreau-Yosida regularization). After illustrated these two tools both in the optimization framework and in the applications to the selection of Nash equilibria in normal-form games, we present a constructive selection method for SPNEs based on the Tikhonov regularization in one-leader N-follower two-stage games (with N=1 and N>1), and a constructive selection method for SPNEs based on a learning approach which has a behavioural interpretation linked to the costs that players face when they deviate from their current actions (relying on the proximal point methods) in one-leader one-follower two-stage games

On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

Modeling several competitive leaders and followers acting in an electricity marketleads to coupled systems of mathematical programs with equilibrium constraints,called equilibrium problems with equilibrium constraints (EPECs). We consider asimplified model for competition in electricity markets under uncertainty of demandin an electricity network as a (stochastic) multi-leader-follower game. First ordernecessary conditions are developed for the corresponding stochastic EPEC based ona result of Outrata [17]. For applying the general result an explicit representation ofthe co-derivative of the normal cone mapping to a polyhedron is derived (Proposition3.2). Later the co-derivative formula is used for verifying constraint qualificationsand for identifying M-stationary solutions of the stochastic EPEC if the demand isrepresented by a finite number of scenarios