Best response functions

The form of the response function to be fitted depends on the goal of modeling, and the amount of available theoretical and experimental information. If we simply want to avoid interpolation in extensive tables or to store and use less numerical data, the model may be a convenient class of functions such as polynomials. In many applications, however, the model is based an theoretical relationships that govern the system, and its parameters have some well defined physical meaning. A model coming from the underlying theory is, however, not necessarily the best response function in parameter estimation, since the limited amount of data may be insufficient to find the parameters with any reasonable accuracy. In such cases simplified models may be preferable, and with the problem of simplifying a nonlinear model we leave the relatively safe waters of mathematical statistics at once. 

We are ready for the first important GT concept best response functions. 

Definition 1. Given an n—player game, player i s best response (function) to the strategies X-i of the other players is the strategy x that maximizes... 

Taken together, the two best response functions form a best response mapping or in the more general case RP RP, Clearly, the best response is the best player i can hope for given the decisions of other players. Naturally, an outcome in which all players choose their best responses is a candidate for the non-cooperative solution. Such an outcome is called a Nash Equilibrium (hereafter NE) of the game. 

The expression says that the slopes of the best response functions are negative, which implies an intuitive result that each player s best response is monotoni-cally decreasing in the other player s strategy. Figure 2.2 presents this result for the symmetric newsvendor game. The equilibrium is located on the intersection of the best responses and we also see that the best responses are, indeed, decreasing. 

Hence, increasing best response functions is the only major requirement for an equilibrium to exist players objectives do not have to be quasi-concave or even continuous. However, to describe an existence theorem with non-continuous payoffs requires the introduction of terms and definitions from lattice theory. As a result, we restricted ourselves to the assumption of continuous payoff functions, and in particular, to twice-differentiable payoff functions. 

A challenge associated with the contraction mapping argument is finding best response functions because in most SC models best responses cannot be found explicitly. Fortunately, Theorem 5 only requires the derivatives of the best response functions, which can be done using the Implicit Function Theorem (from now on, IFT, see Bertsekas 1999). Using the IFT, Theorem 5 can... 

Returning to the newsvendor game example, we have found that the slopes of the best response functions are... 

As with the contraction mapping approach, with two players the Theorem becomes easy to visualize. Suppose we have found best response functions X = fi x2) and X2 = /2( i) as in Figure 2,2. Find an inverse function X2 = fi xi) and construct an auxiliary function g xi) = f xi) — f2 xi) that measures the distance between two best responses. It remains to show that g x ) crosses zero only once since this would directly imply a single crossing point of fi xi) and f2 x2)- Suppose we could show that every time crosses zero, it does so Jrom below. If that is the case, we are assured there is only a single crossing it is impossible for a continuous function to cross zero more than once from below because it would also have to cross zero from above somewhere. It can be shown that the function g xi) crosses zero only from below if the slope of g xi) at the crossing point is positive as follows... 

Intuitively, the first player chooses the best possible point on the second player s best response function. Clearly, the first player can choose a NE, so the leader is always at least as well off as he would be in NE. Hence, if a player were allowed to choose between making moves simultaneously or being a leader in a game with complete information he would always prefer to be the leader. However, if new information is revealed after the leader makes a play, then it is not always advantageous to be the leader. 

Denote the best response function of the retailer by Rr Q) and the best response function of the wholesaler by Ryj A) both defined for zero initial inventory. At this point we have not demonstrated uniqueness or even existence of the equilibrium. It helps, however, to visualize the problem first. We begin by presenting the game graphically (see Figure 14.1, parameters are taken from the example in Section 6 with r = 8). The point A, is a Nash... 

Note that the objective function is concave, and hence the retailer s first-order condition characterizes the unique best response. For stationary policies, it is sufficient to consider the best response functions in the single-period game. The slope of the retailer s best-response function is found by implicit differentiation as follows ... 

Proof The retailer acts second by solving (14.9). The wholesaler takes the retailer s best response function into account and solves the following problem ... 

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... 

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

Definition 2. We call a pair At, sequential-move MISTJZAC game (where the government is the firs mover) if and only if A" = A" andd = argmax u A , d), where the best response function X defined as X = argmax U A,d). 

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