Let us consider a version of the vertical product differentiation model in the tradition of Shaked and Sutton [] and Mussa and Rosen []. We assume a population of consumers who have utility function U = qup if they buy one unit of the differentiated good and U = 0 if they do not buy. The symbols u and p stand for quality and price of the good, and q represents a taste parameter. We assume the distribution of q to be uniform with q Î [0,[`(q)]] and a density S.
We assume that there exist two firms in the industry, A and B. In the first period of the game they decide on the quality they want to produce u_{j} and incur a fixed (i.e. independent of the quantity produced) cost of quality F_{j} = k_{j} [(u_{j}^{2})/ 2], with j = A,B. The cost of quality can be thought of as R&D investments or advertising outlays. The quadratic form we have taken is a standard assumption which greatly simplifies the calculations. Any convex function would give the same qualitative results. Note that the two firms do not necessarily have the same technology: firm A is at least as efficient as firm B, with 1 = k_{A} £ k_{B} = k. Therefore, the parameter k is a measure of the asymmetry existing between the two firms. We take marginal production costs to be constant and, without loss of generality, we set them equal to zero.
In the second and final stage of the game, firms decide on the price at which they want to sell their product. We work as usual by backward induction, and solve the last stage of the game first.
We first find the demand schedules faced by the top and bottom quality firm respectively as:


At the first stage of the game, the net profit functions for the firms are given by p_{1} = P_{1} k_{j}[(u_{1}^{2})/ 2], and p_{2} = P_{2} k_{i} [(u_{2}^{2})/ 2], with j,i = A,B and j ¹ i.
Note that we have deliberately not specified which firm is producing the top and which the bottom quality, since either firm can be the high (low) quality provider at equilibrium. Indeed, there might exist two equilibria in pure strategies (we do not consider mixed strategies here). In the first one, it is the more efficient firm A which produces the top quality. In the second, it is the less efficient firm B. We now turn to the characterization of both equilibria, and to the analysis of their existence.

(1) 

(2) 
By dividing the two equations above, rearranging and writing u_{1} = r u_{2} with r ³ 1 we obtain:

(3) 

where: g(k) = 343+2568k+1920k^{2}+4096k^{3}; h(k) = 686+2967k+3552k^{2}+5888k^{3}.
Note that [(¶p_{2})/( ¶u_{2})] = 0 can be written as: u_{2} = S [`(q)]^{2} [(r^{2}(4r7))/( (4r1)^{3})]. By replacing r with r_{A}, we obtain u_{2}^{*} and u_{1}^{*} = r_{A} u_{2}^{*} as functions of k only (the term S [`(q)]^{2} has just a scale effect throughout).
Figure (lefthand panels) depicts the qualities and profits for this candidate equilibrium E_{1}(u_{1}^{*},u_{2}^{*}), where the top quality is provided by the more efficient firm A. Note that the top quality makes considerably higher profits than the low quality firm.

(4) 

(5) 
Let us write u_{1} = zu_{2} (with z ³ 1) and use the same procedure followed in the previous section to derive the solution. We can then find the value z_{B} which satisfies the firstorder conditions, and by substitution the solution E_{2} = (u_{1}^{**},u_{2}^{**}). For completeness, we report here the value z_{B} which is:

where: l(k) = 4096+1920k+2568k^{2}+343k^{3}; m(k) = 5888+3552k+2967k^{2}+686k^{3}.
Figure 1 (righthand panels) reports qualities and profits at this candidate solution.
Note that the two pairs of candidate solutions have been obtained under the hypothesis that no firm can deviate from the quality it has been assigned. For instance, in the first case the candidate solution E_{1} was found under the hypothesis that firm A produces the top quality, and firm B the bottom quality. But to make sure that the pair (u_{1}^{*},u_{2}^{*}) is really an equilibrium, we also have to check that firm B does not find it profitable to 'leapfrog' the rival and provide a quality higher than (u_{1}^{*}). In other words, it must be checked that there exists no quality u_{1} ¢ such that p_{1} ¢(u_{1} ¢,u_{2} = u_{1}^{*}) ³ p_{2}^{*}(u_{1}^{*},u_{2}^{*}). Likewise, it must be checked that firm A does not have an incentive to deviate by supplying a quality which is lower than u_{2}^{*}. Indeed, it is possible to show that these deviations are not profitable, and therefore conclude that the pair (u_{1}^{*},u_{2}^{*}) is always an equilibrium (see Motta, Thisse and Cabrales, 1997, for an illustration in a similar model). The discussion below should give more insight about this result.
The same exercise must be made for the second case, where firm B produces the top quality. However, it turns out that this is not an equilibrium for all the values of the parameters. Indeed, there exist high enough values of parameter k (to be precise, k = 1.5894 is approximately the threshold value above which this equilibrium collapses) for which the more efficient firm finds it profitable to produce a quality u_{1} ¢ higher than the quality u_{1}^{**} the rival would produce at the candidate solution. In other words, p_{1} ¢(u_{1} ¢,u_{2} = u_{1}^{**}) can be higher than p_{2}^{**}(u_{1}^{**},u_{2}^{**}), as can be seen from figure .
To understand why this equilibrium breaks down when technological asymmetries are large enough, consider the extreme case where firm B is infinitely inefficient. If k tends to infinity, then firm B will choose a top quality u_{1}^{**} = e, with e arbitrarily small, since a huge investment must be made to produce even a very low quality. At the candidate equilibrium, firm B is making infinitesimally small profits, and firm A's profits are even lower (the bottom quality firm always makes less profit). It is then clear that the latter firm has an incentive to deviate from the candidate equilibrium. At a small cost, it can produce a quality higher than e, become the top firm and earn higher profits.
Consider for instance the following variation of the model presented above. Firms have exactly the same technologies (k = 1 for both), but when the game starts the firms have inherited different levels of quality (which can be interpreted as the consequence of past levels of R&D or advertising expenditures). In the first period of the game, they can update the quality of the good by incurring some adjustment costs; in the second period, they compete on prices.
Two equilibria might arise: one where the firm endowed with the larger initial quality will still be producing the higher quality at the new equilibrium (persistence of dominance) and one where it is the initial laggard firm which provides the high quality (leapfrogging); the latter equilibrium ceases to exist when the difference in initial quality levels is too large^{1}. This model has been analyzed, in an international trade context by Motta, Thisse and Cabrales (1997) ^{2}
The tracing procedure is intended to model the reasoning process of the agents in search of a unique prediction of the game. A first step for the tracing procedure is to construct an auxiliary game in which the agents think the others play according to the bicentric priors with some weight and the actual strategy with the remaining weight. The procedure starts with the priors being given all the weight. As the weight to the actual strategy increases, a point may be reached when there is more than one equilibrium, but there is only one that is joined by a continuous curve to the initial equilibrium. This unique equilibrium joined by a continuous curve to the one resulting from the best responses to the bicentric priors is called the risk dominant equilibrium.
We consider a 2 player game, G, where the strategy space for player i = A,B is U_{i} (in our case the qualities), and the payoff function is p_{i}: U_{A} ×U_{B}® Â.
Since the risk dominance criterion assumes that only one of two equilibrium strategy pairs will be played, only strategies that are somehow connected with the equilibrium strategies should be a part of the preliminary expectations. To do this more formally, let us define a game G¢ as a formation of G if the set U_{A}¢×U_{B}¢ formed by reducing the strategy sets from the original game and maintaining the payoff function is closed with respect to best replies in G (that is, all strategies in G¢ are best responses to some other strategy in G). In other words, G¢ is a formation of G if

(6) 
formation^{3} such that u^{*} and u^{**} belong to the strategy sets in F. F will be the game used for the risk dominance comparison between u^{*} and u^{**}.
The preliminary expectations used by Harsanyi and Selten [] are given by the bicentric priors. For every z with 0 £ z £ 1 and for i = 1,2 define r_{i}^{z} as the best response for player i, if she believes that j will play equilibrium v_{j} with probability z and v¢_{j} with probability 1j. If there is more than one best response, the centroid of the set of best responses is selected. The centroid of a set U¢_{i} is a mixed strategy c that gives the same weight to all strategies in the set and none to strategies outside, that is, c(u_{i}) = 1/U¢_{i} if u_{i} Î U¢_{i} and c(u_{i}) = 0 otherwise; where U¢_{i} is the number of elements in U¢_{i}.
Player j does not know what probability player i assigns to v_{j} and v¢_{j}, so player j will compute her bicentric prior by giving the same weight to all possible z for player i (adopting the principle of insufficient reason). So we can write

(7) 
These bicentric priors represent the initial expectations of each player about the other player's strategy. These expectations need not be consistent with the actual strategy the other player intends to play given her preliminary expectations, and thus the best responses to preliminary expectations need not be equilibria. The best responses to preliminary expectations will be used by the tracing procedure to start a path that will smoothly approach one equilibrium which will be the one selected by the risk dominance criterion.
The (linear) tracing procedure is defined as follows: for 0 £ t £ 1, let the game G^{t} be the game with the same strategy sets as F, and whose payoff functions are defined by

(8) 
For any game G^{t} let E^{t} be the set of equilibrium points in G^{t} and let X = X(F,p) the graph of the correspondence t ® E^{t} for 0 £ t £ 1. A continuous path L contained in X connecting point x^{0} = (0,u^{0}) and x^{1} = (1,u^{1}) (where u^{t} is an equilibrium of G^{t}, t = 0,1), is called a feasible path. The linear tracing procedure consists in selecting an equilibrium of a game F which is the strategy part u^{1} of the endpoint (1,u^{1}) of a feasible path L. The linear tracing procedure is well defined if X contains one and only one feasible path. Take a pair (F, p) for which the tracing procedure is well defined and let u^{1} be the equilibrium selected. We denote then T(F, p) = u^{1}.
We say that an equilibrium v risk dominates v¢ if, given a bicentric prior p, and a reduced game F, T(F, p) = v.
We have computed numerically the preliminary expectations and then applied the tracing procedure for a game with payoff functions like the ones in section 2 and strategy spaces reduced to the two equilibrium strategies and one hundred convex combinations of them. We find that for all the parameter values for which we do the computation (k = 1.1, 1.2, 1.3, 1.4, 1.5 and notice that for k > 1.5 there is a unique equilibrium) the equilibrium with the low cost firm being the leader is one selected by risk dominance in all the cases. Table 1 reports the results of the tracing procedure. For selected values of t between 0 and 1^{4} we show the equilibrium values of the auxiliary games. One can easily see that there is only one path connecting the equilibrium of the game with the preliminary expectations and the actual game played. This path is the constant path that already starts at the equilibrium values of quality for leadership by the low cost firm. For sufficiently high values of t another path arises, which finally connects with the other equilibrium. This level of t is higher as the value of k rises. This is similar to the rising value of the degree of risk dominance, [(LA_{0}LB_{0})/( LA_{1}LB_{1}+LA_{0}LB_{0})], we defined for 2×2 games, and can be interpreted as saying that the equilibrium where the low cost firm is the leader is ``more" risk dominant as the cost asymmetry increases.
^{2} See also Cabrales and Motta (1996) for another model with very similar features.
^{3} Such a set exists because the intersection of two formations is a formation.
^{4} We only report 10 values of t, but the computations were done for 100.