What determines the outcome of a price war, and why isn't that outcome reached instantaneously? Mary is making a hefty profit manufacturing and selling widgets. Jim has some money laying around and he is trying to figure out if he shouldn't start manufacturing some widgets too. 

In this example assume that the marginal cost to produce a widget is zero (time, money, etc are all close enough to zero to be indistinguishable), but the cost of making a widget manufacturing plant is quite high. 

Also assume that the market for widgets is fairly centralized. There are two bins with widgets in them and consumers can purchase their widgets from either bin. They cannot be resold, however. There are laws against that for whatever reason, and they are ruthlessly enforced. You may only sell widgets that you manufacture yourself.

What does Jim decide? And if he decides to enter the widget business, what is the eventual price per widget and how long does it take to reach that price?

More (but not too) formally: 
Our situation can be modeled by a few different games,

Case 1: assume the rules of the price war are

players take turns setting a new, lower price from $\mathbb R$ or passing. When both players pass, the prices are locked and the market is allowed to run for t.

in this case the optimal strategy to set your price to whatever price your opponent sets, if the prices are already equal, then pass.

Case 2: 

The market is allowed to run for time t after each player sets a (lesser or equal) price, they take turns setting prices, and they choose prices from $\Bbb Z$.

Here the optimal strategy is in fact no different. if the price starts at $p1$, and the sharing strategy is $S$, then there exists some $t$ such that for all $t'$>$t$ $$EV(S,S,t)=\frac{t*p}{2} &gt;tk(p-1) &gt; EV(S,S',t)$$ for all $p,k,S'$

Case 3: 

Both players set prices in each timestep t without the information of the price their opponent sets in that time step, and the market is run. 

In this case, no pure strategy is guaranteed to exist, because the information is not perfect. I don't know what the Nash equilibrium would end up being, nor do I know that there is any reason to suspect that it converges (as t -> 0) to the same price as the previous two cases, despite the game seeming to do so.  

So I guess the question becomes, why does competition even exist at all in this market, and since all of these scenarios seem to converge towards the reality. ?
Answer to question

If we take your assumptions literally, Jim will decide not to enter the widget business. For suppose he did incur the cost of entry and that Mary is selling at price $p_m$. Jim can only sell to consumers if his price $p_j\leq p_m$. The best price for Jim is $p_m-\epsilon$ (where $\epsilon$ is some very small, positive amount). But this would leave Mary with no sales, so she will have an incentive to reduce her price to $p_j-\epsilon$. This is the price war you describe and it will result in both parties reducing their price to marginal cost (i.e. zero—this is known as Bertrand competition).

Since there are no frictions in your model, this will all happen very quickly. Moreover, because a price of zero implies zero profits, Jim has no incentive to incur the cost of entry in the first place, so he will instead opt to stay out of the market.



Effect of relaxing assumptions

This is obviously quite a stylised result owing to the stark assumptions of the model. But it makes a good foundation for thinking about some more realistic settings. For example:


If Mary's factory has a marginal cost $c_m$ and Jim has a patent on a new technology that implies cost $c_j&lt;c_m$ then Jim can profitably set a price below the lowest price Mary is willing to set. Good news: if someone invents a more efficient technology they can enter the market and displace the older, less efficient technology.
Suppose that the factory has a monthly fixed maintenance cost. If Mary is a small, credit-constrained independent firm and Jim is a huge conglomerate with large reserves of cash then Jim can enter and practice predatory pricing. If he is patient enough, he can enter the market, set $p_j=0$, and wait for Mary to run out of money. She will then leave the market and Jim becomes a profitable monopoly. This kind of anti-competitive behaviour is illegal in most jurisdictions (yes, there are laws against setting too low a price!).
Suppose that Mary's widgets are blue and Jim's are pink. Consumers have an idiosyncratic preference for either blue or pink widgets. Then both firms can set a positive price and sell to the consumers who like their colour more. The more are products differentiated the more there is scope for both firms to exist profitably in the industry and Jim may find it worthwhile to enter. This is why firms talk so much about differentiation and unique selling points. There are various ways to model this in economics. Here is an example.




More formally, $p_j=p_m=0$ is the unique pure strategy Nash equilibrium

To be slightly more formal, let us check that $p_m=p_j=0$ is indeed a (Nash) equilibrium of the subgame in which both firms simultaneously set $p\in\mathbb{R}$ with the lowest priced firm capturing the entire (finite) demand. Neither party can profit by deviating to $p&lt;0$ as this yields negative profits. A deviation to any $p&gt;0$ results in the rival firm having a lower price so demand (and profits) are zero---again not profitable. Thus, there is no profitable deviation and $p_i=p_m=0$ is an equilibrium.

Is there another pure strategy equilibrium with some $p&gt;0$? The answer is no. Consider the three possibilities and observe how a profitable deviation can be constructed for each:


$p_m&gt;p_j$. In this case $j$ could increase his price to some $p_j'\in(p_j,p_m)$ without losing any demand. Such a $p_j'$ exists by the connectedness of the real line.
$p_m=p_j=p$. In this case at least one of the firms must be capturing less than all of the consumers. But by deviating to $p'=p-\epsilon$ it can capture all consumers. If we write $D$ for demand before the deviation and $D'&gt;D$ for demand afterwards then the change in profits is $D'(p-\epsilon)-Dp$. By the connectedness of the real line, there exists an $\epsilon$ sufficiently small that this is positive.
$p_m&lt;p_j$. This case is symmetric to $p_m&gt;p_j$.


Thus, the only pure strategy equilibrium of this one-shot game is $p_j=p_m=0$.



If we repeat the game then we can sustain other (collusive) equilibria

What if we repeat the game infinitely many times? Suppose the two firms have an implicit understanding (an explicit agreement would be illegal) that they will both set $p=p^*$ for some $p^*&gt;0$. Moreover, it is understood that if one firm deviates from this behaviour today, then both firms will revert to playing the static equilibrium ($p_j=p_m=0$) forever. Firms discount the future at rate $\delta$. For simplicity, suppose that demand is constant: $D(p)=1$ (not crucial).

If a firm plays in accordance with this understanding (and expects its rival to do likewise) then its profit is

$$\sum_{t=0}^\infty\frac{1}{2}\delta^t p^*=\frac{p^*}{2(1-\delta)}.$$

If a firm cheats and sets $p^*-\epsilon$ ($\epsilon$ small) then it captures the whole demand today, but expects to be punished forever thereafter, so profit is $p^*$. Thus, both firms wish to comply with their implicit understanding if
$$\frac{p^*}{2(1-\delta)}&gt;p^*\iff \delta&gt;\frac{1}{2}$$
(i.e. if they are sufficiently patient).

So, if we repeat the game and firms are very patient, we can sustain any price in equilibrium. But note that these equilibria require not only that firms are patient, but also that a) there is no threat of future entry that could destabilise the equilibrium; b) firms are able to coordinate on a $p^*$ without ever making an (illegal) explicit agreement; and c) firms are able to constantly and accurately monitor rivals behaviour in order to detect cheating.