2 Bach-Oligopolies and Game Theory

 https://www.youtube.com/watch?v=JMq059SAQXM&ab_channel=JacobClifford

The prisoner's dilemma is the "rubber bone" of game theory-it can  be chewed over forever. Thousands of mathematicians, psychologists,  political scientists, philosophers, and economists have thought about  it, trying to find a solution. Yet it remains as mysterious and baffling 

as in 1950, when Merrill Flood and Melvin Drescher first proposed it.  The name prisoner's dilemma was given by Albert W Tucker, who in  1951 wrote the first paper about it. Tucker presented the problem in  the form of a short detective story. Here is one version: 

Let's summarize the situation in a table:

The first number in each cell, the one in boldface, indicates what the first accomplice gains, while the second number shows the gain of the other accomplice. Since it is worse to be jailed for ten years than for five, years of imprisonment must be considered a negative gain, whence the minus signs. In the given situation the best achievable result is zero, being set free.

Everyday Prisoner's Dilemmas 

month, a state law prohibiting changing prices during the month. The new price has to be posted exactly at midnight on the first day of the month. The owner of one of the stations might think thus: I made a small profit at last month's price, but not much. If the other station were to go out of business, my sales would double, and I would then make a huge profit since the costs of maintaining my station would rise very little. It would be worth a little sacrifice to take customers away from my competitor. What if I lowered my price a little? Although I would make a little less profit on each gallon of gas, my sales would almost double, and this would be well worth it. After complicated calculations, the owner of the gas station ascertains that if he lowered his price a certain amount and thereby won over half of his competitor's customers, then his one unit of profit would rise to four units. But then doubt begins to gnaw at him: What if the owner of the other station also thought this way and also lowered his price? Then my sales wouldn't go up a bit! Worried, he makes new calculations and determines that next month his business would bring him zero profit with the lowered price. Thus, it is not worth lowering the price. Now that doubt has awakened within him, he feels compelled to perform additional calculations: What would happen if he kept his old, higher, price and the other owner lowered his price? The outcome would be disastrous! Maintenance costs are so high that with only half the sales volume, his deficit would be three units even at the higher price. Midnight is approaching, and if he wants to set a new price, he must post it as the clock strikes twelve. Just in case, he has prepared a new board at a lower price. If he sees his neighbor lowering his price, he can quickly do the same. Slowly he walks out to his price board as midnight begins to strike, and he can see that the other owner is walking to his price board, also looking worried, clutching a small board. The two are just about to enter into a conversation when they notice the dreaded Inspector of Municipal Order, who is watching what will happen to the prices at midnight. There is no time for negotiations, which anyhow are illegal. Both owners have to decide immediately whether or not to change the price or leave the old one. The critical moment is at hand-the stroke of midnight and neither can see what the other is doing. 

The situation can again be summarized in a table:

The logic of the arms race also suggests a typical prisoner's dilemma. Balance could develop between the two opposing superpowers either if both sides armed themselves to the teeth or if both of them armed themselves only moderately. The cheaper balance of power is clearly better for both parties than an expensive equilibrium. Now our table looks like this:

The numbers show the order of desirability: A score of 1 represents the worst possible outcome in the situation, while 4 is the best. An expensive balance is better than defenselessness, and superiority is better than a cheap balance. This order of values can be questioned, and it should be, but many believe in it, especially if superiority can be turned into direct economic advantage. Game theory assumes that the players are clearly aware of their own (perceived) interests and values. It is not the task of game theory to change them, but by its very abstractness, game theory can call attention to the necessity of change-for example, when it clearly points out that a certain system of values inevitably leads to a prisoner's dilemma. The prisoner's dilemma is principally about cooperation, its evident necessity and frequent near-impossibility. In all of our examples, one of the strategies involved cooperation, while the other did not. The prisoner who does not confess, the gas station owner who does not lower his price, the superpower that does not arm itself excessively, are cooperative. With this behavior, if both parties think similarly, a better result can be achieved. The noncooperative strategy will be called competitive, although this word does not always express the essence of the thing. It is not quite suitable to apply this term to Tosca.

Prisoner's Dilemmas with Many Persons

The above examples have shown that cooperation usually means giving up something. Thus one can easily find oneself in a prisoner's dilemma. The formula is the following: Take a temptation that, if everybody succumbs to it, leads to catastrophe. But that is not all. A special arrangement of values is also necessary. There are other severe dilemmas to which the conclusions drawn from the prisoner's dilemma cannot be easily applied. The million-dollar game does not function this way, although there was a temptation there, too, and if everybody had succumbed, no one would have profited. The difference lies in the fact that in the prisoner's dilemma, the competitive player causes harm to cooperative players, while in the million-dollar game the competitive player causes harm only to the other competitors. The prisoner's dilemma with many persons is also called the problem of common pastures, which is modeled by the following example: A village has a common pasture. There are ten farmers who have cows, one each, and all ten cows graze on the common pasture. There they wax fat, thereby more or less eating up the meadow. The farmers get richer, and one or two of them can soon afford a second cow. When the first farmer sends his second cow to pasture, little change is noticed. Perhaps slightly less grass can be eaten by anyone cow; perhaps one fatted calf will become less so. When the second and third farmers send their second cows to pasture, still no great problems arise. Although the cows become visibly thinner, each remains well-fed and healthy. But by the time the seventh farmer gets around to buying a second cow, all the animals are hungry, and the total value of the seventeen cows is less than that of the original ten. By the time the ten farmers have two cows each, all twenty cows have starved to death. Throughout this process, two cows are always worth more than one, and so it is always advantageous for a farmer to buy a second cow until they all starve to death. From this description, it is apparent that we have found ourselves once again in the logic of the prisoner's dilemma. But beware, not all social dilemmas are prisoner's dilemmas. Let us have a look at the common pastures "game" table:

Once again, the numbers represent the degree of benefit: The best case scores 4, the worst scores 1. The second numerals in the table indicate how the others fare in the given situation on average. A pre-cise analysis of the game would require a more complex table that shows whether each of the ten farmers is cooperating or competing. That complex table is summarized in this small one, in which the behavior of only one farmer is emphasized. The numbers in the table are precisely those in the arms race table, and therefore the logic here is that of the prisoner's dilemma. This table is valid until all the cows have starved to death. When this happens, the numbers in the table change, but by then it is too late to profit from the knowledge that the prisoner's dilemma has been operating again. Panic is a typical example of a prisoner's dilemma with many persons; when a fire breaks out in a crowded room, for example. There is a special case, one that has often occurred, when the door of the room opens inward. The cooperative behavior would be for everyone to step back a couple of steps, allowing the door to be opened easily, and then everybody would be saved. This generally does not occur: Everyone runs to the door, crushing one another to death.

Iterated Prisoner's Dilemmas

The original story of the prisoner's dilemma describes a particularly critical situation in that 1 have only one choice, after which everything will be over. If I, as one of the accomplices, do not cooperate and my partner makes the mistake of trying to cooperate, then he will not have to reproach me for at least ten years. If he, too, does not

cooperate, then he'll have nothing to reproach me for when we get out of jail in five years. The situation is slightly different when we face the same partner repeatedly. In this case, if we do not cooperate at the outset, we sentence ourselves to eternal competition, because the partner we cheated once will probably never trust us again. The case of the two service station proprietors is an iterated prisoner dilemma, since on the first day of next month they will meet the same dilemma, unless the unilaterally cooperating partner goes bankrupt in the meantime. When watering lawns is prohibited during a drought, it is another case of an iterated prisoner's dilemma. The cooperative behavior is to obey the order, while it is competitive behavior when somebody secretly waters his lawn, achieving personal gain but endangering the whole community's water supply. There are similar examples to be found in relation to pollution of the environment. The logic leading to competition is not complete in a situation with repetition. If there are many rounds, the choice is not only between cooperation and competition. Complex long-term strategies are available as well. For example, one possible strategy is that at first I cooperate, but if my partner does not reciprocate, then I will never cooperate again. Another possible strategy is to cooperate all the time, hoping thereby to make my opponent see reason sooner or later. Or I can cooperate every other time, regardless of my partner's move. The possibilities are infinite. Prisoner's dilemmas do not arise only in human interactions. Sticklebacks exhibit very interesting behavior when a large fish approaches. They cannot tell in advance whether the large fish wants to make a meal of them. It would be simple to flee from every large fish, but then stickleback life would be one of constant flight, and they would have no time for other vital activities. On the other hand, the fatalistic solution, "Let's see what this big fish will do," can also be dangerous, since an unexpected attack could destroy a whole school of sticklebacks. Consequently, the sticklebacks do the following. A reconnaissance group gradually approaches the large fish. They swim toward it, stop for a while, approach, swim a few more inches, stop, approach, and so on. If they get so close that the large fish could easily attack them and nothing untoward happens, they return to the

other sticklebacks and continue their usual activity. If, however, the large fish catches one of the reconnaissance group, then the rest rush back to the others to sound the alarm. The prisoner's dilemma arises within the reconnaissance group. One or two sticklebacks may chicken out and turn tail. Those who return will individually be safe, but if they all flee, then all of the sticklebacks may fall victim to the large fish, including the deserting scouts and their offspring. If the remaining reconnoiters do not flee, they will be more endangered individually than before the few fled, because each one's chances of being caught will have increased if the large fish turns out to be a stickleback eater. The logic of the situation matches exactly that of the problem of common pastures. We will soon see how sticklebacks cope with this dilemma.

Axelrod's Competitions

The American political scientist Robert Axelrod has investigated the problem of whether cooperation can theoretically develop in a world where everybody is governed by his own interests. In 1979, Axelrod asked several well-known scientists, many of whom had published papers on the prisoner's dilemma, to participate in a competition. He asked them to send him the strategy they considered optimal for the iterated prisoner's dilemma. Axelrod asked for the strategy in the form of a computer program. He then had the programs play against one another in a round-robin competition: Each program played a series of two hundred iterations against each of the others. In each of the two hundred rounds, the programs received scores according to the following table: 

The winner of the round-robin competition would be the program with the highest cumulative score. Axelrod did not reveal the number

of rounds in advance so that this information could not be programmed into the competing strategies. Fourteen programs entered the competition, from the very simple to the highly elaborated. The most complex of these attempted to outsmart their opponents using concepts borrowed from the field of artificial intelligence. To these programs, Axelrod added a fifteenth, which cooperated or competed at random. The winning program, that of the famous social psychologist Anatol Rapoport, was the simplest of all. It consisted of just two rules: 1. Cooperate in the first round. 2. Thereafter, do whatever the opponent did in the previous round. Rapoport named his program "Tit for Tat." It is abbreviated TFT in the game-theory literature. What is so ingenious in this embarrassingly simple strategy that it was able to defeat the highly wrought programs of the best experts?