The game model in our application is patterned after similar implementations [10][13][1] of the Green and Porter model for noncooperative implicit collusion under imperfect information.[2] The outcome of this game has also been referred to as a self enforcing cartel agreement, because the game has many equilibria in which the players can choose to cooperate or to not cooperate. Also, because the cooperative choices can be self enforcing equilibria, the participants are behaving as cartel members even though they are oligopolistic competitors. The participants make inferences about the behavior of the other firm by observing the market price. If the market price remains above a certain value, called the trigger price, then the firm does not infer noncooperative behavior on the collusive agreement. But if the market price does drop below the trigger price, then some punishment must be applied.
As in the Green and Porter model, the players select
to produce a certain quantity in each period of the game.
The Pareto optimal equilibria is limited to a set of strategies
where the Cournot quantity is the noncooperative equilibria
that a player can select to produce.
The total market
quantity produced results in a market price, which serves as
information to the firms when they choose a quantity to
produce in the next time period.
In cooperative periods, a smaller quantity, which results in a higher discounted profit, can be selected by players. The interactive game begins in a cooperative period and it would be best for the participants to maintain cooperative behavior to extract the highest price. However, outside forces, such as a temporary flux in demand, a demand shock, or some firm changing their quantity selection, can cause changes in the market price.
If the market price falls below a certain trigger price, then all firms will begin to produce at a higher quantity level. This is easiest to model with only two players, because there is no need to have a mechanism to oversee that punishments are enforced.
In an ``ex post'' evaluation of how this game actually evolves, all players correctly infer that their rival chose the cooperative quantity in the last period, and that price is low because of demand shock. Punishment then follows automatically as a self-enforcing reaction to the low realized demand. Because ex ante: players chose an adequate punishment and stuck to it, regardless of whatever the reason for the price drop, ex post: no cheating takes place. But, it is optimal to punish only because players do not know what caused the drop in price. Under this uncertainty, players cannot abuse the implicit agreement. During punishment periods, players will produce at higher quantities and accept lower prices, because otherwise all firms would have incentive to defect during cooperative periods, and cooperative quantities would never be enacted.
By applying this quantity game we can find the
optimal trigger strategies of the two players.
That is, we look for the 4-tuple, of trigger price (tp),
number of punishment periods (T), non-cooperative equilibria
quantity ( 
), and the response quantity ( 
)
which is the optimal output
of one player in response to the optimal output of the other
player.
The trigger strategy ( 
) for player i
maximizes the player i's expected present and future discounted value.
Similarly, for player j the trigger strategy ( 
)
maximizes the player j's expected present and future discounted value.
So, in our model the expected present discounted value functions for two firms in the market are as follows.
where
That is, F(x) is the proabability that the expected market price (mp) is greater than the trigger price (tp). We use the same cumulative distribution function that was used by Bierman and Fernandez in their model of this quantity game.[1][page 449] This cumulative distribution function has a median and mean of 1, and a standard deviation of 0.071. It was selected for ease of use and also so we could verify our model with the referenced model's example.
Other notation is defined as follows.
and 
are the strictly noncooperative Cournot quantities
that the two firms can choose.
is the discount factor, the way in which firms value future
profits.
, 
are the response quantities which optimize the
value statements.
and 
are profits using the inverse demand
function and costs described earlier.
