Generating Unique Searching Paths

As the above example showed, for an NE to be found with Algorithm 1, one of the players’

equilibrium strategy must be chosen in the forward search. I.e. an equilibrium strategy must be a best response to at least one strategy of the competitor which is chosen in the forward

search. Only then the equilibrium strategies can be chosen as best responses; either directly in the forward search or in the backtracking procedure.

As a counter example consider the game described by Table 5.2. If one of the strategies s¹₁, s¹₂, s²₁ or s²₂ is chosen as the starting strategy in line 3 of our algorithm, the pure NE (s¹₃, s²₃) will not be found by Algorithm 1 because none of the equilibrium strategies is a best response to one of these strategies and hence cannot be chosen in the forward search. In order to find the pure NE, one of the equilibrium strategies would have to be chosen as the starting strategy in Algorithm 1 as it is. In case this starting strategy is not picked, a new one would have to be chosen and the algorithm would have to be rerun. To avoid this, we here extend the algorithm so that it only needs to be started once and finds a pure NE with certainty if one exists.

r¹ s²₁ s²₂ s²₃ Table 5.2: Payoff Matrices for the Exemplary Game

For this, a unique starting strategy must be provided in line 3 of the algorithm to allow the starting player to exploit his full range of booking limit combinations—from non-competitive to competitive. We will explain this in more detail in Section 5.3.5.1. Next, the algorithm continues with line 4. However, if it now becomes necessary to forbid the starting player’s strategy from the first iteration and to seek an alternative strategy, constraint (5.9) is added not with constraint (5.10) but with a constraint that merely makes sure that the starting player’s revenue must not be lower than his non-competitive revenue. This allows the starting player to choose all of his possible strategies in iteration 1 and has the same effect as if the algorithm were rerun repeatedly with these strategies as starting strategy and player 2 as the starting player.

If now the starting player’s model cannot be solved in the first iteration, the game does not have a pure NE. For, in this case all possible starting strategies will have been considered without finding an NE. However, the extended algorithm will still terminate in finite time because the players have only finite numbers of available (integer) strategies.

However, depending on the parameters (demand, capacities, prices etc.) there might be several (pure) NE in a game as the example in the previous section showed. This, on the other hand, would lead to coordination problems about the players’ choice of NE strategies. If both players choose strategies that constitute the same NE, they indeed end up in one. However, if they choose strategies corresponding to different NE, they fail to pick an NE. See also (Fudenberg and Tirole, 1991, Section 1.2.4). In Harsanyi and Selten (1988) a rule for selecting a unique NE was provided.

Furthermore, in order to pick a certain NE, all of them need to be computed first. Finding all of them, on the other hand, might not be possible since degenerated games might have infinite sets of equilibria as noted in Section 4.3. So one can be satisfied with finding only one NE.

An issue that arises with algorithms which terminate after finding only one NE, such as the algorithms by Lemke and Howson (1964) and McKelvey and McLennan (1996), is that they might find different NE in degenerated games depending on the starting strategy and, even more, on the choice of best responses during the iterative search. Practically speaking, different

5.3 Computing an Exact Pure Nash Equilibrium solvers and/or algorithms employed by the competing airlines for solving the models on hand might lead to different optimal solutions (and hence different best responses) if several exist.

Similarly, in (primally) degenerated LPs there may be more than one candidate for entering and/or exiting the basis in a pivot step. Depending on the choice of the variables leaving and entering the basis, one might end up in different optimal solutions of the LP if several exist.

Obviously, the coordination problem with NE disappears if there is only one (pure) NE in the game or if every model has only one unique solution (in which case the iterative search would have a unique searching path). Indeed, the majority of literature mentioned above that investigates competition in an RM setting using game theory, is concerned with deriving conditions under which a game’s NE is unique. However, if these conditions are not fulfilled, the uniqueness is not guaranteed and an algorithm possibly tailored to find the unique NE is useless. The same applies when a player has several best responses to his opponent’s equilibrium strategy and is completely indifferent between them. However, coordination issues might have their roots even earlier, namely in the iterative search for best responses. If a player has several best responses to his competitor’s strategy—i.e. his model has several optimal solutions—and he is completely indifferent about them, the players might end up making different predictions about the other’s behavior if they have different solvers and/or employ different algorithms to solve the models.

For this reason, we extend our algorithm to generate a unique searching path in the course of the iterative search independently of the type or structure of the game, the parameters, or the number of optimal solutions for a model (and thus best responses to a player’s strategy). This makes the algorithm suitable for games where uniqueness of an NE is not guaranteed or cannot be established. For this, we use a perturbed price matrix so that the model turns out to have a unique solution and hence generates unique best responses. Perturbation, first described by Charnes (1952), is a common tool in linear programming for turning degenerated models into a non-degenerated ones. See also e.g. Murty (1983, Ch. 2.5.7–2.5.8). In Lemke and Howson (1964) perturbation was used to turn a degenerated game into a non-degenerated one.

In a degenerated game a model’s candidate optimal solutions (a player’s best responses) express themselves through equal payoffs. However, if several optimal solutions exist for a player’s model M^a, they differ in the booking limits for substitutable (combinations of) products which use the same legs and generate the same revenue. In order to make the optimal solutions unique and to eliminate the possibility of substituting one product for another we add a matrix τ^a (which has the same dimensions as the player’s price matrix) of an exponentiated factor τ^a to a player’s matrix of prices. For this, the prices must be sorted in a fixed order which must not be changed once it is determined. Depending on the order of perturbation, the elements in the matrix τ^a are exponentiated differently. Equation (5.13) shows such a matrix with the perturbation order “first alphabetic by O&D name, then by fare class”. The rows of the matrix stand for the O&D pairs and the columns stand for the fare classes. The second superscripts in the matrix stand for exponents which in this case are computed as “F ∗ (row number −1) + column number”.

Together with the unique starting strategy motivated above, applying perturbation generates unique best responses and thus unique searching paths which leads to both players ending up in the same NE even if several exist. This way, instead of dealing with uniqueness results of NE, we can guide the search for an NE and are able to avoid coordination issues in our own way.

In our case, the information connected to the perturbation matrices can be interpreted as an airline’s long-term goals. If an airline e.g. wants to establish itself as a premium airline, it can perturb the prices so that in case of a tie, the higher-class product is given a priority.

The opposite is the case if the airline pursues a diversified customer structure. Such strategic goals are made public e.g. in the companies’ annual reports so that competitors can deduce the order of perturbation from each other’s strategic goals. Hence, this order can be thought of as common knowledge drawn from the airlines’ long-term goals which can be interpreted as implicit communication between the players and resolve coordination issues (Holler and Illing, 2009, p.

86). What is important, is that the assumption of complete information can be maintained given the perturbation. We could also model the implications connected to the perturbation explicitly through constraints but more constraints might lead to lower payoffs and the model still can have several optimal solutions. Instead, the order in which the price matrix is perturbed can be used to control the choice of the optimal solution and make sure that it is unique.

Comparing our algorithm with the algorithm by Audet et al. (2001), we can see that the latter is less target-oriented since it tries almost all combinations of strategies to see whether they constitute an equilibrium. While our Algorithm 5.1 searches for best responses in every step, the algorithm by Audet et al. (2001) allows the modification of only one player’s model in each step (node of the search tree). Furthermore, while our algorithm uses a backtracking procedure in case of repeating best responses which do not constitute an NE, their algorithm does not suppress this possibility and actively adds certain constraints to the models repeatedly.

Also, with our algorithm, we cannot receive infeasible models in the forward search which can happen with the algorithm for enumerating all extreme NE. Finally, while we satisfice with a single NE and propose a perturbation of the players’ prices to deal with degenerated games, Audet et al. (2001) explicitly developed their algorithm to find all NE in degenerated games as well.

5.3.5 Computational Study