The Optional Prisoner's Dilemma (OPD) game models a situation of conflict involving two players in game theory. It can be seen as an extension of the standard prisoner's dilemma game, where players have the option to "reject the deal", that is, to abstain from playing the game.^{[1]} This type of game can be used as a model for a number of real world situations in which agents are afforded the third option of abstaining from a game interaction such as an election. ^{[2]}
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✪ Optional calculus proof to show that MR has twice slope of demand  Khan Academy

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Transcription
For those of you who are curious and have a little bit of a background in calculus, I thought I would do a very optional and when I say it's optional, you don't have to understand this in order to progress with the economics playlist, but a very optional proof showing you that in general, the slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve, assuming that the demand curve is a line. So if the demand curve looks like that, this is price. This is quantity. That is demand right over there. I'm going to show that the marginal revenue curve has twice the slope. It is twice as steep as this. It's really twice the negative slope. So let me just write price as a function of quantity. We know we get price. Since this is a line, we can essentially write it in our traditional slope yintercept form. In algebra class, you would write if this was y and this was x, you would write y = mx+b where m is the slope and b is the yintercept. I'll do something very similar, but instead of y and x we have P and Q. So, if P = m x Q where m is the slope plus ... plus the Pintercept + b. So this right over here is b and if you were to ... If you were to take your, If you were to take your change in P, if you were to take your change in P and divide it by your change in Q ... your change in Q, you would get m. That is your slope, change in P / change in Q. Now what is going to be our total revenue? And this is kind of ... We're just kind of almost doing what we've done in the last few videos, but we're doing it in general terms. So if this is total revenue, total revenue as a function of quantity. Well, total revenue is just price x quantity. Total revenue is just price x quantity. We've already written ... We've already written price as a function of quantity right over here, so we could take that and substitute it ... and substitute it in right over there. So we get total revenue is equal to and I'll write it all in blue. We have mQ + b and then we're going to multiply that x Q. We're going to multiply that x Q or we get total revenue = mQ^2. mQ^2 + b x Q And this is a parabola and it's actually going to be a downward sloping parabola because m is going to be negative. This is downward sloping. m has a negative slope, so m is negative. So we know ... We know that M < 0 over here. That's one of the assumptions we'll make. If m < 0, this is going to be a downward opening parabola. Total revenue will look something, total revenue will look something like that. That is our total revenue. Now, the marginal revenue as a function of quantity is just the derivative and this is the calculus part. It's the slope of the tangent line at any given point and that is what the derivative is. It's the slope of the tangent line at any point as a function as a function of quantity. So you give me a quantity, I will tell you what the slope of the tangent line of the total revenue function is at that point. So we essentially just have to take the derivative of this with respect to Q. So we get D, TR/DQ. So how much does total revenue change with a very, very small change in quantity, infinitely small, infinitesimal change in quantity and this comes straight out of calculus. m is a constant. Q^2, the derivative of Q^2 with respect to Q is 2Q. So it's going to be 2Q x the constant. So it's going to be 2m ... 2mQ. And then b is a constant. We're assuming it's given. b is a constant. The derivative of bQ with respect to Q is just going to be b. It's just going to be b. And so right over here, this is our marginal revenue curve. or I should say our marginal line. It is 2mQ + b. So notice, it has the same yintercept as our demand curve so definitely starts right over there, but it has twice the slope. The slope of our demand curve is m. The slope of our marginal revenue curve is 2m, is 2m and this is a negative slope, so this will be twice as negative. So it will look something like this. It will look something like this, just like that. So no matter what your demand curve is, if you assume it's a line like this, the marginal revenue curve will be a line with twice the slope and in this case, it's twice the negative slope which is kind of ... what's going to be generally true. Anyway, if you understood that, great. You now feel good that this is always the case for a linear demand curve like this. If you did not understand it, don't worry. You can proceed with the economics playlist. You don't need to know calculus for this playlist.
Payoff matrix
The structure of the Optional Prisoner's Dilemma can be generalized from the standard prisoner's dilemma game setting. In this way, suppose that the two players are represented by the colors, red and blue, and that each player chooses to "Cooperate", "Defect" or "Abstain". ^{[3]}
The payoff matrix for the game is shown below:
Cooperate  Defect  Abstain  

Cooperate  R, R  S, T  L, L 
Defect  T, S  P, P  L, L 
Abstain  L, L  L, L  L, L 
 If both players cooperate, they both receive the reward R for mutual cooperation.
 If both players defect, they both receive the punishment payoff P.
 If Blue defects while Red cooperates, then Blue receives the temptation payoff T, while Red receives the "sucker's" payoff, S.
 Similarly, if Blue cooperates while Red defects, then Blue receives the sucker's payoff S, while Red receives the temptation payoff T.
 If one or both players abstain, both receive the loner's payoff L.
The following condition must hold for the payoffs:
T > R > L > P > S
References
 ^ Cardinot, Marcos; Gibbons, Maud; O'Riordan, Colm; Griffith, Josephine (2016). "Simulation of an Optional Strategy in the Prisoner's Dilemma in Spatial and Nonspatial Environments". From Animals to Animats 14. Lecture Notes in Computer Science. 9825. pp. 145–156. doi:10.1007/9783319434889_14. ISBN 9783319434872.
 ^ Batali, John; Kitcher, Philip (1995). "Evolution of altriusm in optional and compulsory games" (PDF). Journal of Theoretical Biology. 175 (2): 161–171. doi:10.1006/jtbi.1995.0128.
 ^ Cardinot, Marcos; O'Riordan, Colm; Griffith, Josephine (2016). "The Optional Prisoner's Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights". ECTA. Proceedings of the 8th International Joint Conference on Computational Intelligence. 1. pp. 86–93. doi:10.5220/0006053900860093. ISBN 9789897582011.