Stackelberg equilibrium

In a Stackelberg equilibrium, the firms in the industry compete on quantity in a sequential fashion. One firm moves first, and then each moves in turn until the last. Working by backwards induction, the last firm seeks to maximize its profits, given the output decisions of all the other firms. The Stackelberg model was rejected because it is more complicated to implement and in most, if not all, of the markets being represented, there were no firms with a leading market share. 

To find an equilibrium of a Stackelberg game which often called the Stackelberg equilibrium we need to solve a dynamic multi-period problem via backwards induction. We will focus on a two-period problem for analytical convenience. First, find the solution X2 x ) for the second player as a response to any decision made by the first player ... 

For having an Increasing Failure Rate (IFR) distribution, there exists a unique Stackelberg equilibrium solution pair A ) in the single-period... 

Since the retailer s best response function is single-valued, the Stackelberg equilibrium exists. Further, we will show that the second derivative of the wholesaler s objective function is negative (i.e., the objective function is concave), and so the Stackelberg equilibrium is unique. The first derivative of the wholesaler s objective function is... 

Proposition 5. There is a unique Stackelberg equilibrium solution pair in the single-period game with a powerful retailer and without starting inventory that is characterized by the following system of equations ... 

Since the wholesaler s best response function is single-valued, the Stackelberg equilibrium exists. The first derivative is... 

The retailer s objective function is clearly concave, and the uniqueness of the Stackelberg equilibrium follows. Finally, the optimality conditions are found by equating the first derivatives to zero. 

Two French workers, Villard and de Forcrand, were the most prolific researchers of the period before 1934, with over four decades each of heroic effort. Villard (1888) first determined the existence of methane, ethane, and propane hydrates, de Forcrand (1902) tabulated equilibrium temperatures at 1 atm for 15 components, including those of natural gas, with the exception of iso-butane, first measured by von Stackelberg and Muller (1954). 

Cachon and Zipkin (1990), we consider games where each player chooses a stationary policy the retailer s strategy is A and the wholesaler s strategy is order-up-to quantity Q, Three situations arise either the wholesaler or the retailer might have negotiation power in the supply chain and act as Stackelberg leaders, or it is possible that the players have equal power and therefore the solution is in the form of a Nash equilibrium. We will consider all three situations. 

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