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# A game theory model of the free market

1. ## A game theory model of the free market

Section 1:

If you know how to read a strategic form game and a prisoners' dilemma game, skip this section. Refer to Figure A for this section. All Figures are at the bottom of post.

The boxes represent the choices made by each player. For example, the top left is when both players choose Snitch. The bottom left is when Row chooses Don't Snitch and Column chooses Snitch. Choices are made simultaneously.

The numbers indicate payoffs to each player for each box. With Row choosing Don't Snitch and Column choosing Snitch, payoff for Row is -1 and payoff for Column is 3. The higher the number, the higher the payoff. Payoff structure accounts for every bit of preference in decisions made.

In this game, the Nash equilibrium is top left. Nash equilibrium means that no player would have chosen a different move given what the other player(s) chose. This game type, prisoners' dilemma, is very famous and is a unique type of game because it has a Nash equilibrium where the players are worse off than if they had chosen different particular moves. The stars represent the choices each player makes given the choice of the other player. The double stars means Nash equilibrium.

An example illustrating why the prisoners' dilemma has this payoff structure: two suspects of a crime (who did it) are being interrogated by the cops and the cops are trying to get each to snitch on the other. If one snitches and the other doesn't, it is the best outcome for the snitcher and worst for the snitchee (snitcher goes free, snitchee gets a tremendous amount of jail time). If neither snitch, it is a good outcome for both but not the best (both get small jail time). If both snitch, it is bad for both but not the worst (both get a good deal of jail time but not a tremendous amount). A preliminary analysis of the situation suggests that the players would be best off if they choose to not snitch, since that results in only a small amount of jail for each. But that ultimately isn't the case. The Nash is where they both snitch.

Section 2:

If you understand the more generalized form of the prisoners' dilemma and how the Nash arises through the gameplay, skip this part. Refer to Figure B for this section.

An initial idea for how the prisoners' dilemma might not have a Nash where each player is worse off is that both players can coordinate actions. They could discuss actions before playing the game and agree to coordinate. In that case they would avoid the Nash equilibrium of the worse payoff. But this doesn't work since when one has credibility on coordinating, the other receives better payoff by defecting. So, the next question is how does this work if they play the game multiple times, surely they would coordinate then. It makes sense, but when we do rollback equilibrium, we see that the players will still not choose to coordinate. In the real world, a player might choose to coordinate early on (essentially by mistake) but then will swiftly choose to defect.

The rollback equilibrium is discovered as follows. Let's say this game will be played 5 times. The best outcome for Row is for both to choose coordinate 4 times then for Row to defect on the 5th play while Column coordinates. The same goes for Column but vice versa. Since Column knows Row will defect on the 5th play, Column will defect on the 4th play. They will both then defect on subsequent plays. But since Row knows that Column will defect on the 4th play, Row will defect on the 3rd play. And since Column knows Row will defect on the 3rd play, Column will defect on the 2nd play. This goes all the way down to the very first play, where each player's best option is to defect immediately and for the stretch of the game, leaving them worse off than if they had coordinated (though they would not coordinate even if given the option).

Figures C graph the time and payoff for Row if he defects at t=4, t=3, t=2 while Column coordinates until after Row defects.

Section 3:

No more skipping!

Does this mean the prisoners' dilemma is unsolvable? No. One way to approach solving it is to make an unspecified number of plays. In that case, neither player knows when it would be best for them to defect, so instead they just keep coordinating, as seen in Figure D. But there is a problem, as seen in Figure E. If the gain from defecting early on is high enough that it covers all the lower payoff for the rest of the play for the game, then even if the game lasts for infinity, players will choose to defect as soon as they can. A sufficient gain from one-time defecting can come if the payoff is high enough that the remaining time playing the game doesn't reach it. This can happen via payoffs being in terms of money, which means that interest rates apply and at a sufficiently high payoff of one defect and a sufficiently high interest rate, the growth will exponentially surpass any of the linear gains from the rest of the game. The shaded area in Figure E represents what portion would need to be a big enough payoff for a player to choose it. Figure E doesn't show the exact amount that would be needed, just what portion of the graph would need to have large enough size in order to incentivize defection.

Section 4:

Now for the free market.

Figure D is essentially the Invisible Hand observation by Adam Smith, the philosopher from whom the field of economics spawned. The Invisible Hand is the observation that when producers compete freely for consumers (and consumers are free to choose), the self-interest of each individual producer aligns with the self-interest of the consumers. This is because when the self-interest doesn't align regarding one producer, that producer loses his consumers on the margins, who then go to other producers. This results in the non-aligning producer in having marginal losses and thus changing to align with the consumer or going bankrupt. Adam Smith described this competition of self-interest as being like an invisible hand that makes the society better even though each producer (and each consumer) is pursuing their own self-interest.

Figure D models one player's actions compared to another player's, as presented in previous diagrams. It can be applied to multiple players playing multiple games. But as stated above, Figure D doesn't represent reality if the payoff of one defect is high enough (which it most certainly would be some of the time).

Because of what is shown in Figure E, a common response is to have government intervene into the market in an attempt to make it so that there is not a sufficiently high payoff to a one-time defect that would deter players from choosing coordinate. I won't comment on how effective that idea is or how effective it has been, though I do want to comment on how the free market could adjust for the problem presented in Figure E.

The idea is arbitration in order for a player to enter the game in the first place. It would be along the lines of each player purchases bonds that the arbitrator controls that, if the player were to defect, the bonds would be paid to other players in the game. If these bonds are large enough, then they would eliminate the incentive to defect. In the particular game presented, a bond that costs the player 3 if the player defects while the other coordinates would be sufficient enough to keep a player from defecting. Bonds could mature on a periodic basis, and players who wish to continue in the game could purchase new arbitration/bonds at each required time.