Business competition is not that kind of trick that allows you to cheat on your opponent once and run away. Dishonesty can be either punished or tolerated by your opponent when the next competition arrives. Given that, you and your opponent will compete with each other for a looooooooong time, what can you say about your strategies? Would you like to be honest and cooperative? ... or, dishonest and deceptive?
Take your time, play around with the simulator below.
> We modified the value of price and demand function: Price is now \(p(q)=30-q\) and Cost is now \(c(q)=2\).
You may take this as an exercise, use previous knowledges to calculate your optimal quantity to produce, or start with any value you like!
> We use \(q_i\) denote the amount to produce, \(\pi_i^k\) denote the profit in the kth round, and \(\pi_i\) denote the accumulated profits.
> This calculator only accept integer values, this assumption will be explained later.
> This game consists real-time plot of profits and accumulated profits, if you cannot see the graph on the right hand side properly, please refresh the page.
This is Round 1
Spending a while on the above calculator, you may find that it is tricky to identify any valuable pattern or strategies. This is reasonable, after all, when the game is repeated many times, the game itself is destined to be more difficult to analyze.
However, the payoff matrix in the last section somewhat motivates us to conclude two profound strategies in one time Cournot Competition.
Strategy I: Cooperate:
With collusion, both firms will have an optimal amount to produce that will maximize their joint payoff.
To cooperate means to produce such amount of products.
Strategy II: Cheat:
As the cooperation is not stable, one can unilaterally maximize his payoff under the assumption that his opponent is cooperative.
We once again turn to this familiar matrix. This time, we'll regard this competition as a two-person game where both players have two strategies, cooperate (Producing \(q^c\)) or cheat (Producing \(q^b\))
| Firm 1 \ Firm 2 | Cooperate (Produce 7) | Betray (Produce 10.5) |
|---|---|---|
| Cooperate (Produce 7) | \((98, 98)\) | \((110.25, 73.5)\) |
| Betray (Produce 10.5) | \((73.5, 110.25)\) | \((73.5, 73.5)\) |
Hence, we impose two new assumptions into this scenario:
| Firm 1 \ Firm 2 | Cooperate (Produce \(7\)) | Betray (Produce \(10\)) |
|---|---|---|
| Cooperate (Produce \(7\)) | \((98, 98)\) | \((110,77 )\) |
| Betray (Produce \(10\)) | \((77, 110)\) | \((80, 80)\) |
| Prisoner 1 \ Prisoner 2 | Cooperate (Not Confess) | Betray (Confess) |
|---|---|---|
| Cooperate (Not Confess) | \((R,R)\) | \((S,T)\) |
| Betray (Confess) | \((T,S)\) | \((P,P)\) |
We'll play the infinitely cournot duopoly game once again, but this time, you choose a strategy between cooperation and cheat rather than the exact output value. Play around with it! See if you can find anything interesting.
> There are two players (each with a different strategy), you may choose one of them to compete with.
> For this game, you only need to choose either 0 (Cooperate) or 1 (Cheat) as your strategy to enter into the input field.
> We use \(q_i\) denote the amount to produce, \(\pi_i^k\) denote the profit in the kth round, and \(\pi_i\) denote the accumulated profits.
> This calculator only accept integer values, this assumption will be explained later.
> This game consists real-time plot of profits and accumulated profits, if you cannot see the graph on the right hand side properly, please refresh the page.
Choose a player you want to play with!
You've chosen Player as your opponent.
Unable to change in the middle, click restart.
This is Round 1
Grim Trigger
Player starts by fully cooperating with his opponent. However, he never cooperates once he got betrayed.
Carrot and Stick
Player starts by fully cooperating with his opponent. However, he will betray back for a fixed number of times, \(T\), if he once got betrayed.
When the punishment ends, the player will continue to fully cooperate. When \(T=1\), this strategy is also as known as tit-for-tat.
Definition of \(T\) can be extended to any positive number. For example, if \(T=\dfrac{1}{2}\), this means the player will punish once when he was betrayed twice.
Aggressive Rate
Strategies like Grim Trigger and Carrot and Stick are rather a protective mechanism,
however, if you don't cheat (being aggressive), we've seen that you cannot gain anymore than the mutual cooperation result.
Therefore, if a player want to gain more, maybe he should cheat actively to attempt obtaining a higher payoff.
We'll assume that a player will attack (to cheat) if his opponent cooperates in a row for \(A\) times, and this defines the aggresive rate.
We provide you with a sandbox where you can freely manipulate the protective mechanism and aggressive rate (as defined in the last section). It is your time to determine whether to be conservative, aggressive, or both!
> T, protective mechanism, is the number of punishment on your opponent once you've got betrayed.
> A, aggressive rate, is the number of times your opponent cooperates in a row that you attack once.
> This game contains a real-time animated plot, if you cannot see it properly, please refresh.
> If the animation accidentally stops, click the "start the competition" button again to continue.
Your input is invalid, check!!!
For those who are interested, we'd like to introduce one more kind of strategy named Zero-Determinant Strategy. One may seek the paper "Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent" by Press, W. H. and Dyson, F. J. for more theoretical details. A hyperlink is provided in the reference section of the credit page.
Long story short, zero-determinant strategies is a kind of strategy applied to all games taking the form of Iterated Prisoners' Dilemma that forces linear relationship between two players' payoff. Surprisingly, as the thesis points out:
This kind of strategy takes a special form, as it is based on the outcome of the previous round. We'll state the optimal strategies and final payoff without proof. First thing first, for an Iterated Prisonners' Dilemma, we have such a payoff matrix:
| Player 1 \ Player 2 | Cooperate (c) | Betray (d) |
|---|---|---|
| Cooperate (c) | \((R,R)\) | \((S,T)\) |
| Betray (d) | \((T,S)\) | \((P,P)\) |
We know
Then a zero-determinant strategy with extortionate factor \(\chi\) is just $$\tilde{\mathbf{p}}=\phi\left[(\mathbf{S}_x-P\mathbf{1})-\chi\left(\mathbf{S}_y-P\mathbf{1}\right)\right]$$ where \(0 \le \phi \le \dfrac{P-S}{(P-S)+\chi(T-P)}\) (WLOG, we shall assume \(\phi\) to be the midpoint among this interval throughout the later discussion).
Also, it can be shown that under this strategy, the payoff for both players are $$s_x=\dfrac{P(T-R)+\chi\left[R(T-S)-P(T-R)\right]}{(T-R)+\chi(R-S)}$$ and $$s_y=\dfrac{\chi P(R-S)+\left[R(T-P)-S(R-P)\right]}{(T-R)+\chi(R-S)}$$ where \(s_x\) is the payoff of the player using zero-determinant strategy, and \(s_y\) is the payoff of his opponent.
These are rather abstract symbols that is not intuitive at all. To sum up, these figures and symbols basically tell us:
Despite that we won't prove all these results, but the extortionate factor itself is already interesting enough to explore. Try to freely alter the extortionate factor, take a look at both players' payoffs.
> Note that \(T = 110\), \(R = 98\), \(P = 80\), and \(S = 77\).
> We restrict the extortionate factor's range to 0 to 10. Recall that an extortion too high will make the extortion target lose incentive to cooperate.
If the other player decides to accept the extortion, then:
Extortionate Profit:
Extorted Profit:
After traversing all that about the Cournot Competition, backtracking to the very beginning, we're sure that you've already got some clues why the protection scheme imposed by American government actually benefits the Japanese automobiles companies. This is also why the World Trade Organization (WTO) called it an end on Volutnary Export Restraints (VER) in 1994.
Cournot Competition is actually a kind of competition based on homogenous products' competition, this does not only apply to the commercial field, all these techniques you've learnt can also be used on modelling real-life scenarios. We encourage you to model similar scenarios as we've done, into a iterated Prisoners' Dilemma, this not only reduces the difficulty of analyzing such games, but also more intuitive and realistic.
In terms of applications, we provided a calculator that allows you to compute all those solution concepts discussed throughout this website.
Lastly, we once again thanks for whoever exploring this website, hope that you will find this topic interesting and useful for you life! See you!