- Game Theory Problem With 100 Couples And Cheating Relationships
- Game Theory Problem With 100 Couples And Cheating Couples
- Game Theory Problem With 100 Couples And Cheating Stories
14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Normal-Form Games. Once upon a time, there was a village with 100 married couples. The women had to pass a logic exam before being allowed to marry. The high priestess was not required to. No one would ever inform a woman that her husband is cheating on her. A firm might threaten to invoke a trigger in hopes that the threat will forestall any cheating by its rivals. Game theory has proved to be an enormously fruitful approach to the analysis of a wide range of problems. Corporations use it to map out strategies and to anticipate rivals’ responses. The blissfully married couples who survived infidelity: Partners reveal how they forgave their other halves. However, the couple admitted there was.
On test day for my Behavioral Ecology class at UCLA, I walked into the classroom bearing an impossibly difficult exam. Rather than being neatly arranged in alternate rows with pen or pencil in hand, my students sat in one tight group, with notes and books and laptops open and available. They were poised to share each other’s thoughts and to copy the best answers. As I distributed the tests, the students began to talk and write. All of this would normally be called cheating. But it was completely okay by me.
Below: Peter Nonacs talks to KCRW about letting his class “cheat.”
Who in their right mind would condone and encourage cheating among UCLA juniors and seniors? Perhaps someone with the idea that concepts in animal behavior can be taught by making their students live those concepts.
Animals and their behavior have been my passions since my Kentucky boyhood, and I strive to nurture this love for nature in my students. Who isn’t amazed and entertained by videos of crafty animals, like Betty the tool-making crow, bending wires into hooks to retrieve baskets containing delicious mealworms? (And then hiding her rewards from a lummox of a mate who never works, but is all too good at purloining hard-won rewards of others?)
Nevertheless, I’m a realist. Almost none of my students will go on to be “me” – a university professor who makes a living observing animals. The vast majority take my classes as a prelude to medical, dental, pharmacy, or vet school. Still, I want my students to walk away with something more than, “Animals are cool.” I want them to leave my class thinking like behavioral ecologists.
Much of evolution and natural selection can be summarized in three short words: “Life is games.” In any game, the object is to win—be that defined as leaving the most genes in the next generation, getting the best grade on a midterm, or successfully inculcating critical thinking into your students. An entire field of study, Game Theory, is devoted to mathematically describing the games that nature plays. Games can determine why ant colonies do what they do, how viruses evolve to exploit hosts, or how human societies organize and function.
So last quarter I had an intriguing thought while preparing my Game Theory lectures. Tests are really just measures of how the Education Game is proceeding. Professors test to measure their success at teaching, and students take tests in order to get a good grade. Might these goals be maximized simultaneously? What if I let the students write their own rules for the test-taking game? Allow them to do everything we would normally call cheating?
A week before the test, I told my class that the Game Theory exam would be insanely hard—far harder than any that had established my rep as a hard prof. But as recompense, for this one time only, students could cheat. They could bring and use anything or anyone they liked, including animal behavior experts. (Richard Dawkins in town? Bring him!) They could surf the Web. They could talk to each other or call friends who’d taken the course before. They could offer me bribes. (I wouldn’t take them, but neither would I report it to the Dean.) Only violations of state or federal criminal law such as kidnapping my dog, blackmail, or threats of violence were out of bounds.
![Game Game](https://plato.stanford.edu/entries/game-theory/figure13.jpg)
Gasps filled the room. The students sputtered. They fretted. This must be a joke. I couldn’t possibly mean it. What, they asked, is the catch?
“None,” I replied, “You are UCLA students. The brightest of the bright. Let’s see what you can accomplish when you have no restrictions, and the only thing that matters is getting the best answer possible.”
Once the shock wore off, they got sophisticated. In discussion section, they speculated, organized, and plotted. What would be the test’s payoff matrix? Would cooperation be rewarded or counter-productive? Would a large group work better, or smaller subgroups with specified tasks? What about “scroungers” who didn’t study but were planning to parasitize everyone else’s hard work? How much reciprocity would be demanded in order to share benefits? Was the test going to play out like a dog-eat-dog “Hunger Games”? In short, the students spent the entire week living Game Theory. It transformed a class where many did not even speak to each other into a coherent whole focused on a single task – beating their crazy professor’s nefarious scheme.
On the day of the hour-long test they faced a single question: “If evolution through natural selection is a game, what are the players, teams, rules, objectives and outcomes?” One student immediately ran to the chalkboard, and she began to organize the outputs for each question section. The class divided tasks. They debated. They worked on hypotheses. Weak ones were rejected, promising ones were developed. Supportive evidence was added. A schedule was established for writing the consensus answers. (I remained in the room, hoping someone would ask me for my answers, because I had several enigmatic clues to divulge. But nobody thought that far afield!) As the test progressed, the majority (whom I shall call the “Mob”) decided to share one set of answers. Individuals within the Mob took turns writing paragraphs and they all signed an author sheet to share the common grade. Three out of the 27 students opted out (I’ll call them the “Lone Wolves”). Although the Wolves listened and contributed to discussions, they preferred their individual variants over the Mob’s joint answer.
In the end, the students learned what social insects like ants and termites have known for hundreds of millions of years. To win at some games, cooperation is better than competition. Unity that arises through a diversity of opinion is stronger than any solitary competitor.
But did the students themselves realize this? To see, I presented the class with one last evil wrinkle two days later, after the test was graded but not yet returned. They had a choice, I said. Option A: They could get the test back and have it count toward their final grade. Option B: I would—sight unseen—shred the entire test. Poof, the grade would disappear as if it had never happened. But Option B meant they would never see their results; they would never know if their answers were correct.
“Oh, my, can we think about this for a couple of days?” they begged. No, I answered. More heated discussion followed. It was soon apparent that everyone had felt good about the process and their overall answers. The students unanimously chose to keep the test. Once again, the unity that arose through a diversity of opinion was right. The shared grade for the Mob was 20 percent higher than the averages on my previous, more normal, midterms. Among the Lone Wolves, one scored higher than the Mob, one about the same, and one scored lower.
Is the take-home message, then, that cheating is good? Well…no. Although by conventional test-taking rules, the students were cheating, they actually weren’t in this case. Instead, they were changing their goal in the Education Game from “Get a higher grade than my classmates” to “Get to the best answer.” This also required them to make new rules for test taking. Obviously, when you make the rules there is no reason to cheat. Furthermore, being the rule-makers let students behave in a way that makes us a quintessentially unique species. We recognize when we are in a game, and more so than just playing along, we always try to bend the rules to our advantage.
Morally, of course, games can be tricky. Theory predicts that outcomes are often not to the betterment of the group or society. Nevertheless, this case had an interesting result. When the students got carte blanche to set the rules, altruism and cooperation won the day. How unlike a “normal” test where all students are solitary competitors and teachers guard against any cheating! What my class showed was a very “human” trait: the ability to align what is “good for me” with what is “good for all” within the evolutionary games of our choosing.
In the end, the students achieved their goal – they earned an excellent grade. I also achieved my goal – I got them to spend a week thinking like behavioral ecologists. As a group they learned Game Theory better than any of my previous classes. In educational lingo, there’s a new phrase, “flipping the classroom,” meaning students are expected to prepare to come to class not for a lecture, but for a question-and-answer discussion. What I did was “flip the test.” Students were given all the intellectual tools beforehand and then, for an hour, they had to use them to generate well-reasoned answers to difficult questions.
The best tests will not only find out what students know but also stimulate thinking in novel ways. This is much more than regurgitating memorized facts. The test itself becomes a learning experience – where the very act of taking it leads to a deeper understanding of the subject.
Peter Nonacs is a Professor in the Ecology and Evolutionary Biology Department at UCLA. He studies the evolution of social behavior across species, ranging from viruses, to insects, to mammals and even occasionally humans. He wrote this for Zócalo Public Square.
Game Theory and Oligopoly Behavior
Oligopoly presents a problem in which decision makers must select strategies by taking into account the responses of their rivals, which they cannot know for sure in advance. The Start Up feature at the beginning of this module suggested the uncertainty eBay faces as it considers the possibility of competition from Google. A choice based on the recognition that the actions of others will affect the outcome of the choice and that takes these possible actions into account is called a strategic choice. Game theory is an analytical approach through which strategic choices can be assessed.
Among the strategic choices available to an oligopoly firm are pricing choices, marketing strategies, and product-development efforts. An airline’s decision to raise or lower its fares—or to leave them unchanged—is a strategic choice. The other airlines’ decision to match or ignore their rival’s price decision is also a strategic choice. IBM boosted its share in the highly competitive personal computer market in large part because a strategic product-development strategy accelerated the firm’s introduction of new products.
Once a firm implements a strategic decision, there will be an outcome. The outcome of a strategic decision is called a payoff. In general, the payoff in an oligopoly game is the change in economic profit to each firm. The firm’s payoff depends partly on the strategic choice it makes and partly on the strategic choices of its rivals. Some firms in the airline industry, for example, raised their fares in 2005, expecting to enjoy increased profits as a result. They changed their strategic choices when other airlines chose to slash their fares, and all firms ended up with a payoff of lower profits—many went into bankruptcy.
We shall use two applications to examine the basic concepts of game theory. The first examines a classic game theory problem called the prisoners’ dilemma. The second deals with strategic choices by two firms in a duopoly.
The Prisoners’ Dilemma
Suppose a local district attorney (DA) is certain that two individuals, Frankie and Johnny, have committed a burglary, but she has no evidence that would be admissible in court.
The DA arrests the two. On being searched, each is discovered to have a small amount of cocaine. The DA now has a sure conviction on a possession of cocaine charge, but she will get a conviction on the burglary charge only if at least one of the prisoners confesses and implicates the other.
The DA decides on a strategy designed to elicit confessions. She separates the two prisoners and then offers each the following deal: “If you confess and your partner doesn’t, you will get the minimum sentence of one year in jail on the possession and burglary charges. If you both confess, your sentence will be three years in jail. If your partner confesses and you do not, the plea bargain is off and you will get six years in prison. If neither of you confesses, you will each get two years in prison on the drug charge.”
The two prisoners each face a dilemma; they can choose to confess or not confess. Because the prisoners are separated, they cannot plot a joint strategy. Each must make a strategic choice in isolation.
The outcomes of these strategic choices, as outlined by the DA, depend on the strategic choice made by the other prisoner. The payoff matrix for this game is given in Figure 11.6 “Payoff Matrix for the Prisoners’ Dilemma”. The two rows represent Frankie’s strategic choices; she may confess or not confess. The two columns represent Johnny’s strategic choices; he may confess or not confess. There are four possible outcomes: Frankie and Johnny both confess (cell A), Frankie confesses but Johnny does not (cell B), Frankie does not confess but Johnny does (cell C), and neither Frankie nor Johnny confesses (cell D). The portion at the lower left in each cell shows Frankie’s payoff; the shaded portion at the upper right shows Johnny’s payoff.
Figure 11.6 Payoff Matrix for the Prisoners’ Dilemma. The four cells represent each of the possible outcomes of the prisoners’ game.
If Johnny confesses, Frankie’s best choice is to confess—she will get a three-year sentence rather than the six-year sentence she would get if she did not confess. If Johnny does not confess, Frankie’s best strategy is still to confess—she will get a one-year rather than a two-year sentence. Gta vice city game cheat code download. In this game, Frankie’s best strategy is to confess, regardless of what Johnny does. When a player’s best strategy is the same regardless of the action of the other player, that strategy is said to be a dominant strategy. Frankie’s dominant strategy is to confess to the burglary.
For Johnny, the best strategy to follow, if Frankie confesses, is to confess. The best strategy to follow if Frankie does not confess is also to confess. Confessing is a dominant strategy for Johnny as well. A game in which there is a dominant strategy for each player is called a dominant strategy equilibrium. Here, the dominant strategy equilibrium is for both prisoners to confess; the payoff will be given by cell A in the payoff matrix.
From the point of view of the two prisoners together, a payoff in cell D would have been preferable. Had they both denied participation in the robbery, their combined sentence would have been four years in prison—two years each. Indeed, cell D offers the lowest combined prison time of any of the outcomes in the payoff matrix. But because the prisoners cannot communicate, each is likely to make a strategic choice that results in a more costly outcome. Of course, the outcome of the game depends on the way the payoff matrix is structured.
Repeated Oligopoly Games
The prisoners’ dilemma was played once, by two players. The players were given a payoff matrix; each could make one choice, and the game ended after the first round of choices.
The real world of oligopoly has as many players as there are firms in the industry. They play round after round: a firm raises its price, another firm introduces a new product, the first firm cuts its price, a third firm introduces a new marketing strategy, and so on. An oligopoly game is a bit like a baseball game with an unlimited number of innings—one firm may come out ahead after one round, but another will emerge on top another day. In the computer industry game, the introduction of personal computers changed the rules. IBM, which had won the mainframe game quite handily, struggles to keep up in a world in which rivals continue to slash prices and improve quality.
Oligopoly games may have more than two players, so the games are more complex, but this does not change their basic structure. The fact that the games are repeated introduces new strategic considerations. A player must consider not just the ways in which its choices will affect its rivals now, but how its choices will affect them in the future as well.
We will keep the game simple, however, and consider a duopoly game. The two firms have colluded, either tacitly or overtly, to create a monopoly solution. As long as each player upholds the agreement, the two firms will earn the maximum economic profit possible in the enterprise.
There will, however, be a powerful incentive for each firm to cheat. The monopoly solution may generate the maximum economic profit possible for the two firms combined, but what if one firm captures some of the other firm’s profit? Suppose, for example, that two equipment rental firms, Quick Rent and Speedy Rent, operate in a community. Given the economies of scale in the business and the size of the community, it is not likely that another firm will enter. Each firm has about half the market, and they have agreed to charge the prices that would be chosen if the two combined as a single firm. Each earns economic profits of $20,000 per month.
Quick and Speedy could cheat on their arrangement in several ways. One of the firms could slash prices, introduce a new line of rental products, or launch an advertising blitz. This approach would not be likely to increase the total profitability of the two firms, but if one firm could take the other by surprise, it might profit at the expense of its rival, at least for a while.
We will focus on the strategy of cutting prices, which we will call a strategy of cheating on the duopoly agreement. The alternative is not to cheat on the agreement. Cheating increases a firm’s profits if its rival does not respond. Figure 11.7 “To Cheat or Not to Cheat: Game Theory in Oligopoly” shows the payoff matrix facing the two firms at a particular time. As in the prisoners’ dilemma matrix, the four cells list the payoffs for the two firms. If neither firm cheats (cell D), profits remain unchanged.
Figure 11.7 To Cheat or Not to Cheat: Game Theory in Oligopoly.
Two rental firms, Quick Rent and Speedy Rent, operate in a duopoly market. They have colluded in the past, achieving a monopoly solution. Cutting prices means cheating on the arrangement; not cheating means maintaining current prices. The payoffs are changes in monthly profits, in thousands of dollars. If neither firm cheats, then neither firm’s profits will change. In this game, cheating is a dominant strategy equilibrium.
Armor games king of towers cheat. This game has a dominant strategy equilibrium. Quick’s preferred strategy, regardless of what Speedy does, is to cheat. Speedy’s best strategy, regardless of what Quick does, is to cheat. The result is that the two firms will select a strategy that lowers their combined profits!
Game Theory Problem With 100 Couples And Cheating Relationships
Quick Rent and Speedy Rent face an unpleasant dilemma. They want to maximize profit, yet each is likely to choose a strategy inconsistent with that goal. If they continue the game as it now exists, each will continue to cut prices, eventually driving prices down to the point where price equals average total cost (presumably, the price-cutting will stop there). But that would leave the two firms with zero economic profits.
Both firms have an interest in maintaining the status quo of their collusive agreement. Overt collusion is one device through which the monopoly outcome may be maintained, but that is illegal. One way for the firms to encourage each other not to cheat is to use a tit-for-tat strategy. In a tit-for-tat strategy a firm responds to cheating by cheating, and it responds to cooperative behavior by cooperating. As each firm learns that its rival will respond to cheating by cheating, and to cooperation by cooperating, cheating on agreements becomes less and less likely.
![Problem Problem](https://textarchive.ru/images/1226/2450555/239f2f71.png)
Still another way firms may seek to force rivals to behave cooperatively rather than competitively is to use a trigger strategy, in which a firm makes clear that it is willing and able to respond to cheating by permanently revoking an agreement. A firm might, for example, make a credible threat to cut prices down to the level of average total cost—and leave them there—in response to any price-cutting by a rival. A trigger strategy is calculated to impose huge costs on any firm that cheats—and on the firm that threatens to invoke the trigger. A firm might threaten to invoke a trigger in hopes that the threat will forestall any cheating by its rivals.
Game Theory Problem With 100 Couples And Cheating Couples
Game theory has proved to be an enormously fruitful approach to the analysis of a wide range of problems. Corporations use it to map out strategies and to anticipate rivals’ responses. Governments use it in developing foreign-policy strategies. Military leaders play war games on computers using the basic ideas of game theory. Any situation in which rivals make strategic choices to which competitors will respond can be assessed using game theory analysis.
One rather chilly application of game theory analysis can be found in the period of the Cold War when the United States and the former Soviet Union maintained a nuclear weapons policy that was described by the acronym MAD, which stood for mutually assured destruction. Both countries had enough nuclear weapons to destroy the other several times over, and each threatened to launch sufficient nuclear weapons to destroy the other country if the other country launched a nuclear attack against it or any of its allies. On its face, the MAD doctrine seems, well, mad. It was, after all, a commitment by each nation to respond to any nuclear attack with a counterattack that many scientists expected would end human life on earth. As crazy as it seemed, however, it worked. For 40 years, the two nations did not go to war. While the collapse of the Soviet Union in 1991 ended the need for a MAD doctrine, during the time that the two countries were rivals, MAD was a very effective trigger indeed.
Of course, the ending of the Cold War has not produced the ending of a nuclear threat. Several nations now have nuclear weapons. The threat that Iran will introduce nuclear weapons, given its stated commitment to destroy the state of Israel, suggests that the possibility of nuclear war still haunts the world community.
Self Check: Game Theory
Answer the question(s) below to see how well you understand the topics covered in the previous section. This short quiz does not count toward your grade in the class, and you can retake it an unlimited number of times.
Game Theory Problem With 100 Couples And Cheating Stories
You’ll have more success on the Self Check if you’ve completed the two Readings in this section.
Use this quiz to check your understanding and decide whether to (1) study the previous section further or (2) move on to the next section.