I have been trying to model a Pay What you Want scheme. Now, for a monopoly it is clear that monopoly will have more profit by setting monopoly pricing assuming all players are selfish. But consider a scenario of 2 sellers one posting a fix price and other a PWYW strategy. In a market of selfish players the PWYW will make huge losses and market would move on to the fix price seller who can now act as a monopoly. So does this imply that there is an incentive for the selfish players to keep PWYW firm in market ? Also, how do I extend this idea to more than 2 firms ?
So does this imply that there is an incentive for the selfish players to keep PWYW firm in market ?
No, unfortunately not. In general equilibrium models (as @FooBar already mentioned) buyer's do not think about the influence of their actions on the seller. Therefore, they would just buy at the lowest possible price.
However, if you have a game theoretic model including uncertainty about profits of the PWYW firm and risk-averse consumers who prefer the PWYW firm for some reason you might get Nash-equilibria in which PWYW firms have positive profits.
Another way to model PWYW firms is to include non-monetary benefits from paying. For example a disutility for being seen as a cheapskate or for exploiting the firm or a positive utility from "doing the right thing" (i.e., paying the price that they think is appropriate). The selfish, rational players will pay a positive price and will generate profits for the firm.
I think in reality the impact of non-monetary benefits is much higher than the impact of asymmetric information with risk-aversion.
Also, how do I extend this idea to more than 2 firms ?
I don't really think it is really interesting to extend this to more than two firms (especially without having solved this for 2 firms), but for both modeling options presented it should be "easy enough".
Not it is not. You're assuming that each selfish player actually has enough market power to keep PWYW firms in the market. Usually, we rather think about buyers as atomistic - each individual's purchase decision has no impact on the seller, which is why they usually do not take into account these externalities.
If you had a monopolistic consumer however, he would pay the PWYW firm exactly the marginal cost: Just enough to keep the firm alive, but no surpluses what-so-ever. The opposite of a monopolistic firm, if you will.
However, I don't think that selfish and rational preferences will get you a solution that gives rise to PWYW firms: As I just hinted at, even if the buyers had market powers, the firm wouldn't make profits, and should just go for the standard oligopoly pricing strategy. The latter weakly dominates the former in all market sizes, and strictly dominates in markets with positive market power.