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  1. 1. University of Hong Kong ECON2101 Chiu Additional materials on game theory The essence of a game of strategy is the interdependence of the player’s decisions. These interactions arise in two ways. The first is sequential in which players make alternating moves. Each player, when it is his turn, must look ahead to how his current actions will affect the future actions of others, and his own future actions in turn. The second kind of interaction is simultaneous in which players act at the same time, in ignorance of the others’ current actions. However, each must be aware that there are other active players, who in turn are similarly aware, and so on. When you find yourself playing a strategic game, you must determine whether the interaction is simultaneous or sequential. Some games such as football have elements of both. A game in general consists of three elements: (1) players; (2) a set of strategies for each player; and (3) a payoff function for each player. All of the above information is commonly known among the players. An equilibrium is a complete description/prescription of actions chosen for all players in such a way that given those prescribed actions by other players, no player will have a unilateral incentive to deviate from his/her prescribed actions. This notion of equilibrium was defined and was proved to exist for a great class of games by John Nash in 1950. Therefore, such an equilibrium concept is commonly known as Nash equilibrium. John Nash received the Nobel prize in economics together with two other game theorists in 1995. Simultaneous Games Game 1: Over-Cover Warfare Every week, Next magazine and Oriental weekly compete to have the most eye-catching cover story. A dramatic or interesting cover will attract the attention of potential buyers at newsstands. Thus every week the editors of Next magazine meet behind closed doors to select their cover story. They do so with the knowledge that the editors of Oriental are meeting elsewhere, also behind closed doors, to select their cover. The editors of Oriental in turn know that the editors of Next are making a similar decision, those of Next know that those of Oriental know and so on. In the competition between Next and Oriental, think of a hypothetical week that produces two major news stories: the 911 story and HK’s economic situation. The editors’ choice of cover story is primarily based on what will attract the most newsstand buyers (subscribers buy the magazine whatever the cover is). Of these newsstand buyers, suppose 70 percent are interested in the impeachment of 911 story and 30 percent in HK’s economic situation. These people will buy the magazine only if the story that interests them appears on the cover, if both magazines have the same story, the group interested in it splits equally between them. What will the two magazines’ decisions? Oriental 911 HK Next 911 35, 35 70, 30 HK 30, 70 15, 15 A strategy is a dominant strategy for a player if that strategy gives a higher payoff than other strategies of his do, irrespective of the other player’s strategy. Games in which each side has a dominant strategy are the simplest games from the strategic perspective. There is strategic interaction, but with a foregone conclusion. The Nash equilibrium where both parties adopt their 1
  2. 2. dominant strategies are called dominant strategy equilibrium. Note that all dominant strategy equilibria must be Nash equilibria, but not vice versa. Sometimes one player has a dominant strategy but the other does not. Games in which only one side has a dominant strategy are also very simple. Why? We can sum up the lessons of these examples into a rule for behavior in games with simultaneous moves. Rule 1: If you have a dominant strategy, use it. Rule 2: Eliminate any dominated strategies from consideration, and go on doing so successively. Game 2: Prisoners’ Dilemma Two suspects were prosecuted for robbing a bank. Now they are under investigation separately and simultaneously. What will each of these suspects do to minimize his year of imprisonment? suspect 2 confess not to confess suspect 1 confess 10,10 0,20 not to confess 20,0 1,1 Note: the number in each cell indicates the year of imprisonment. Game 2a: Another version of Prisoners’ Dilemma Tchaikovsky confess not to confess composer confess 10,10 0,20 not to confess 20,0 1,1 Game 3: Warfare between the mainland and Taiwan (purely fictitious) B A C E D F G H I The grid above shows the positions and the choices of the combatants. A Taiwanese ship at the point I is about to fire a missile, intending to hit a Chinese ship at A. The missile’s path is programmed to hit at the launch; it can travel in a straight line, or make sharp right angled turns every 20 seconds. If the Taiwanese missile flew in a straight line from I to A, Chinese missile defenses could counter such a trajectory very easily. Therefore, the Taiwanese will try a path with some zigzags. All such paths that can reach A from I lie along the grid shown. Each length like IF equals the distance the missile can travel in 20 seconds. 2
  3. 3. The Chinese ship’s radar will detect the launch of the incoming Taiwanese missile, and the computer will instantly launch an antimissile. The antimissile travels at the same speed as the Taiwanese missile, and can make similar 90-degree turns. So the anti-missile’s path can also be set along the same grid starting at A. However, to allow for enough explosives to ensure a damaging open-air blast, the antimissile has only enough fuel to last one minute, so it can travel just three segments (e.g. A to B, B to C, and C to F, which we write as ABCF). If, before or at the end of the minute, one antimissile meets the incoming missile, it will explode and neutralize the threat. Otherwise the missile will go on to hit the Chinese ship. The question is: How should the trajectories of the two missiles be chosen? T1- T2- T3- T4- T5- T6- T7- T8- IFCB IFEB IFED IFEH IHGD IHED IHEB IHEF C1-ABCF H O O O O O O H C2-ABEF O H H H O H H H C3- O H H H O H H H ABEH C4- O H H H H H H H ABED C5- O O O H H O O O ADGH C6- O H H H O H H H ADEH C7-ADEF O H H H O H H H C8- H H H H O H H H ADEB Equilibrium strategies When all simplifications based on dominant and dominated strategies have been used, the game is at its irreducible minimum level of complexity and the problem of the circular reasoning must be confronted head-on. What is best for you depends on what is best for your opponent and vice versa. Here we introduce the concept of equilibrium, or Nash equilibrium in honor of John Nash who developed the concept. An equilibrium is a combination of strategies in which each player’s action is the best response to that of the other. Given what the other is doing, neither wants to change his own move. Rule 3: Having exhausted the simple avenues of looking for dominant strategies or ruling out dominated ones, the next thing to do is to look for an equilibrium of the game. Remark: For some games, a pure strategy equilibrium may not exist (e.g., the game of paper- rock-scissors). In that case, we have to appeal to mixed strategies, for almost all games, a mixed strategy equilibrium always exists. Game 4: Collusion among players of unequal size An important feature of OPEC is that its members are of unequal size. Saudi Arabia is potentially a much larger producer than any of the others. Do large and small members of a cartel have different incentives to cheat? We keep matters simple by looking at just one small country, say Kuwait. Suppose that in a cooperative condition, Kuwait would produce 1 million barrels per day, and Saudi Arabia would produce 4. For each, cheating means producing 1 million extra barrels a day. So Kuwait's choices are 1 and 2; Saudi Arabia's, 4 and 5. Depending on the decisions, total output on the market can be 5, 6, 3
  4. 4. or 7. Suppose the corresponding profit margins (price minus production cost per barrel) would be $16, $12, and $8 respectively. This leads to the following profit table. In each box, the bottom left number is the Saudi profit, and the top right number is the Kuwaiti profit, each measured in millions of dollars per day. Profits (Millions of Dollars/Day) for Saudi Arabia, Kuwait Kuwait production Saudi 1 2 Arabia 4 64, 16 48, 24 Production 5 60, 12 40, 16 Kuwait has a dominant strategy: cheat by producing 2. Saudi Arabia also has a dominant strategy, but this is the cooperative output level of 4. The Saudis cooperate even though Kuwait cheats. The prisoners' dilemma has vanished. Why? Game 5: Output Decisions in Duopoly • there are two firms producing the same product • each firm can produce 1, 2, or 3 units of output at a zero marginal cost • the (inverse) market demand function is as follows Quantity demanded Price 2 16 3 10 4 6 5 3 6 1.5 • How much should each firm produce to maximize their joint profits? • How much will each firm produce based on individual’s incentive? Construct a payoff matrix to answer the above two questions. Sequential Games The general principle for sequential-move games is that each player should figure out the other players’ future responses, and use them in calculating his own best current move. So important is this idea that it is worth codifying into a basic rule of strategic behavior. Rule 4: Look ahead and reason back. Anticipate where your initial decisions will ultimately lead, and use this information to calculate your best choice. Most strategic situations involve a longer sequence of decisions with several alternatives at each, and mere verbal reasoning cannot keep track of them. Successful application of the rule of looking ahead and reasoning back needs a better visual aid. A “tree diagram” of the choices in the game is one such aid. We can get to HKU campus from TST via different routes. We can show them schematically: Admiralty KCR station take a bus HKU Take MTR 4
  5. 5. take a ferry TST take a minibus HKU Central take a taxi HKU This road map, which describes one’s options at each junction, looks like a tree with its successively emerging branches--hence the term “decision tree.” We can use such a tree to depict the choices in a game of strategy, but one new element enters the picture. A game has two or more players. At various branching points along the tree, it may be the turn of different players to make the decision. A person making a choice at an earlier point must look ahead, not just to his own future choices, but to those of others. To remind you of the difference, we will call a tree showing the decision sequence in a game of strategy a game tree, reserving the term decision tree for situations in which just one person is involved. Game 6: Suppose the market for some good is dominated by an incumbent, and a new firm is deciding whether to enter the market. We show the structure of moves and payoffs in the following game tree: Accommodate $100,000 to New Firm Enter Incumbent Fight Price War -$200,000 to New Firm New Firm $0 to New Firm Keep Out What should the new firm do? It has to assign how likely the incumbent will accommodate or make a war. Such probabilities come from the new firm’s beliefs about the incumbent’s profits in each of these cases. Such information is added to the game tree. What will be the outcome? Accommodate $100,000 to New Firm Enter $100,000 to Incumbent Incumbent Fight Price War -$200,000 to New Firm New Firm -$100,000 to Incumbent 5
  6. 6. $0 to New Firm Keep Out $300,000 to Incumbent Traditonally, an sequential game is represented as a normal (more correctly, strategic) form. A strategy of a player thus is a complete description of what a player will do under whatever conceivable scenarios of the game. The above sequential game is thus represented by the following normal form. Note that there are two Nash equilibria in this normal form game: (Enter, accomodate) and (keep out, fight price war). However, some thought tells us that giving that new firm has already entered, it is not in the interest of the incumbent to fight price war because, given that situation, not fighting price war is more profitable than fighting. Hence, we should rule out (keep out, fight price war) as an equilibrium because fighting price war is not a credible threat. incumbent accommodate fight price war enter (100k,100k) (-200k,-100k) new firm keep out (0,300k) (0,300k) Remark: For sequential games, an equilibrium consists of instructions regarding players’ prescribed actions even for the point which is never reached should the equilibrium strategies are followed. It is reasonable to impose a condition for an equilibrium: even at points where are never reached the agents at those points are still required to act optimally even if the equilibrium does not prescribe the point to be reached at positive possibility. Equilibria survive such a refinement is called subgame perfect. Subgame perfect equilibria rule out incredible threats. Game 7. Son and mother: incredible commitment & moral hazard To illustrate the importance of credibility of commitment, let us consider the following game between a son and his mother. 6
  7. 7. Consider the following contract between a son and his mon to make sure that the son will go to sleep at a right time. "The son goes to sleep by 10pm. In this case, he will receive no penalty. Otherwise, he will be shot to death by his mon.'' Suppose the HK law allows this contract to be legal, and suppose the pair arrange video recording of the son's activities so that the court can verify when he goes to bed. Thus we have neither asymmetric information nor difficulty of verification. As long as the son believes that the contract will be enforced, he will go to bed on time. So far so good. But will the mon really want to kill the son when the latter does not abide by the contract? Definitely not. In case the son doesn't go to bed by 10pm, it would be in both the mon and the son's interest to renegotiate the contract. Because mon cannot commit to killing his son when the circumstance arises, the son knows it and will simply ignore the contract. This commitment problem is also known as a time consistency problem. The said contract is time inconsistent: At the outset, mon wants the court to enforce the contract, but after the bad scenario occurs, she rather hopes that the court will not enforce the contract. A time consistent contract is one in which under every contingency there does not exist any alternative to the actions prescribed in contract that makes both parties better off (relative to enforcement of the contract). The above mentioned time inconsistency problem is also called moral hazard. Suppose a billionare knows that his business partner in a southeast country is the son of the president of that country. Then he may think that in case the business were unsuccessful, the president will come out to rescue the firm. Hence, doing business with that business partner is a sure win, and this leads to excessive investment and imprudential rescues. This whole sequence of irresponsible actions nowadays are usually called moral hazard. (More will be said about it in the next topic.) Game 8: to move or not to move This form of moral hazard pointed out right before is closely related to government policy where government is one party of a contract and citizens (or firms) are another party. I have recently read of an interesting quote that illustrates this neatly: "There are two types of countries in the world: those with deposit insurance, and those not knowing they have deposit insurance.'' The following game illustrates the tension between a government and its citizens. Suppose residents near Yellow River are more vulnerable to attack of typhoons. Two states of nature: good state and bad state with probability 0.9 and 0.1. In good state: loss to each resident near Yellow River is zero, but in bad state, the loss is $1m. Residency elsewhere no loss under any state. Suppose the moving cost = $50k , which is less than the expected loss of 0.1x$1m=100k. However, in case of bad state, the government can step in to reduce the resident’s loss to $200k at a cost of $500k Draw a game tree between a resident and a government and find the subgame prefect equilibrium of it. (The resident's decision is to move or not to move. The government's decision is whether to rescue if the resident has decided not to move and the bad state has occurred. Consider the corresponding normal form game and find all Nash equilibria. Game 9: Auctions (Optional, Omitted) English auction---The auction begins with an opening bid. Given knowledge of the opening bid, the auctioneer asks if anyone is willing to pay a higher price. The bids continue to rise until no other participants wish to increase the bid. The highest bidder---the only bidder left---pays the auctioneer his or his bid and takes possession of the item. Second-price, sealed-bid auction---In such an auction, bidders submit bids without knowledge of the bids submitted by others. The person submitting the highest bid wins, but has to pay only the amount bid by the second highest bidder. In both types of auctions, the winner will be the same person and has to pay the second highest valuation of the machine. Show that bidding your true valuation of the item is your dominant strategy for second- price, sealed-bid auction. 7
  8. 8. Game 10: Judicial Procedures (Difficult !!!) There are three alternative procedures to determine the outcome of a criminal court case. Each has its merits, and you might want to choose among them based on some underlying principles. 1. Status Quo: First determine innocence or guilt, then if guilty consider the appropriate punishment. 2. Roman Tradition: After hearing the evidence, start with the most serious punishment and work down the list. First decide if the death penalty should be imposed for this case. If not, then decide whether a life sentence is justified. If, after proceeding down the list, no sentence is imposed, then the defendant is acquitted. 3. Mandatory Sentencing: First specify the sentence for the crime. Then determine whether the defendant should be convicted. The difference between these systems is only one of agenda: what gets decided first. To illustrate how important this can be, we consider a case with only three possible outcomes: the death penalty, life imprisonment, and acquittal. The defendant’s fate rests in the hands of three judges. Their decision is determined by a majority vote. This is particularly useful since the three judges are deeply divided. Judge A holds that the defendant is guilty and should be given the maximum possible sentence. This judge seeks to impose the death penalty. Life imprisonment is her second choice and acquittal is her worst outcome. Judge B also believes that the defendant is guilty. However, this judge unyieldingly opposes the death penalty. Her most preferred outcome is life imprisonment. The precedent of imposing a death sentence is sufficiently troublesome that she would prefer to see the defendant acquitted rather than executed y the state. Judge C holds that the defendant is innocent, and thus seeks acquittal. She is on the other side of the fence from the second judge, believing that life in prison is a fate worse than death. (On this the defendant concurs.) Consequently, if acquittal fails, her second-best outcome would be to see the defendant sentenced to death. Life in prison would be the worst outcome. Judge A’s ranking Judge B’s ranking Judge C’s ranking Best Death Sentence Life in Prison Acquittal Middle Life in Prison Acquittal Death Sentence Worst Acquittal Death Sentence Life in Prison The difference between these systems is only one of agenda: what gets decided first. To illustrate how important this can be, we consider a case with only three possible outcomes: the death penalty, life imprisonment, and acquittal. The defendant’s fate rests in the hands of three judges. Their decision is determined by a majority vote. This is particularly useful since the three judges are deeply divided. What will the judges do under different procedures? 8