Briefly discuss the table and the topic. finitely repeatedgameGames in which playersdo not know when thegame will end; andgames in which playersknow when it will end.FINITELY REPEATED GAMESSo far we have considered two extremes: games that are played only once and games that areplayed infinitely many times. This section summarizes important implications of games thatare repeated a finite number of times, that is, games that eventually end. We will consider twoclasses of finitely repeated games: (1) games in which players do not know when the gamewill end and (2) games in which players know when it will end.Games with an Uncertain Final PeriodSuppose two duopolists repeatedly play the pricing game in Table 10-9 until their prod-ucts become obsolete, at which point the game ends. Thus, we are considering a finitelyrepeated game. Suppose the firms do not know the exact date at which their products willTable 10-9A Pricing Game That Is Finitely RepeatedFirm APriceLowHighLow0,0-40, 50Firm BHigh50,-4010, 10CheatFirmA = $50Managerial Economics and Business Strategybecome obsolete. Thus, there is uncertainty regarding the final period in which the gamewill be played.Suppose the probability that the game will end after a given play is 0, where 0

Question

Briefly discuss the table and the topic. finitely repeatedgameGames in which playersdo not know when thegame will end; andgames in which playersknow when it will end.FINITELY REPEATED GAMESSo far we have considered two extremes: games that are played only once and games that areplayed infinitely many times. This section summarizes important implications of games thatare repeated a finite number of times, that is, games that eventually end. We will consider twoclasses of finitely repeated games: (1) games in which players do not know when the gamewill end and (2) games in which players know when it will end.Games with an Uncertain Final PeriodSuppose two duopolists repeatedly play the pricing game in Table 10-9 until their prod-ucts become obsolete, at which point the game ends. Thus, we are considering a finitelyrepeated game. Suppose the firms do not know the exact date at which their products willTable 10-9A Pricing Game That Is Finitely RepeatedFirm APriceLowHighLow0,0-40, 50Firm BHigh50,-4010, 10CheatFirmA = $50Managerial Economics and Business Strategybecome obsolete. Thus, there is uncertainty regarding the final period in which the gamewill be played.Suppose the probability that the game will end after a given play is 0, where 0 < 0 < 1.Thus, when a firm makes today's pricing decision, there is a chance that the game will beplayed again tomorrow; if the game is played again tomorrow, there is a chance that it will beplayed again the next day; and so on. For example, if 0 = 1/2, there is a 50-50 chance thegame will end after one play, a 1/4 chance it will end after two plays, a 1/8 chance that it willend after three plays, or, more generally, a ()' chance that the game will end after t plays ofthe game. It is as if a coin is flipped at the end of every play of the game, and if the coin comesup heads, the game terminates. The game terminates after t plays if the first heads occurs aftert consecutive tosses of the coin.It turns out that when there is uncertainty regarding precisely when the game will end, thefinitely repeated game in Table 10-9 exactly mirrors our analysis of infinitely repeated games.To see why, suppose the firms adopt trigger strategies, whereby each agrees to charge a highprice provided the other has not charged a low price in any previous period. If a firm deviatesby charging a low price, the other firm will &#34;punish&#34; it by charging a low price until the gameends. For simplicity, let us assume the interest rate is zero so that the firms do not discountfuture profits.Given such trigger strategies, does firm A have an incentive to cheat by charging a lowprice? If A cheats by charging a low price when B charges a high price, A's profits are $50today but zero in all remaining periods of the game. This is because cheating today &#34;triggers&#34;firm B to charge a low price in all future periods, and the best A can do in these periods is toearn $0. Thus, if firm A cheats today, it earnsregardless of whether the game ends after one play, two plays, or whenever.If firm A does not cheat, it earns $10 today. In addition, there is a probability of 1 - 0 thatthe game will be played again, in which case the firm will earn another $10. There is also aprobability of (1 - 0)² that the game will not terminate after two plays, in which case A willearn yet another $10. Carrying out this reasoning for all possible dates at which the gameterminates, we see that firm A can expect to earnIICOPPA = 10 + (1 - 0)10 + (1 - 0)²10 + (1 - 0)³10 + ..100if it does not cheat. In this equation, 0 is the probability the game will terminate after one play.Notice that when 0 = 1, firm A is certain the game will end after one play; in this case, A'sprofits if it cooperates are $10. But if 0 < 1, the probability the game will end after one playis less than 1 (there is a chance they will play again), and the profits of cooperating are greaterthan $10.325 finitely repeatedgameGames in which playersdo not know when thegame will end; andgames in which playersknow when it will end.FINITELY REPEATED GAMESSo far we have considered two extremes: games that are played only once and games that areplayed infinitely many times. This section summarizes important implications of games thatare repeated a finite number of times, that is, games that eventually end. We will consider twoclasses of finitely repeated games: (1) games in which players do not know when the gamewill end and (2) games in which players know when it will end.Games with an Uncertain Final PeriodSuppose two duopolists repeatedly play the pricing game in Table 10-9 until their prod-ucts become obsolete, at which point the game ends. Thus, we are considering a finitelyrepeated game. Suppose the firms do not know the exact date at which their products willTable 10-9A Pricing Game That Is Finitely RepeatedFirm APriceLowHighLow0,0-40, 50Firm BHigh50,-4010, 10CheatFirmA = $50Managerial Economics and Business Strategybecome obsolete. Thus, there is uncertainty regarding the final period in which the gamewill be played.Suppose the probability that the game will end after a given play is 0, where 0 < 0 < 1.Thus, when a firm makes today's pricing decision, there is a chance that the game will beplayed again tomorrow; if the game is played again tomorrow, there is a chance that it will beplayed again the next day; and so on. For example, if 0 = 1/2, there is a 50-50 chance thegame will end after one play, a 1/4 chance it will end after two plays, a 1/8 chance that it willend after three plays, or, more generally, a ()' chance that the game will end after t plays ofthe game. It is as if a coin is flipped at the end of every play of the game, and if the coin comesup heads, the game terminates. The game terminates after t plays if the first heads occurs aftert consecutive tosses of the coin.It turns out that when there is uncertainty regarding precisely when the game will end, thefinitely repeated game in Table 10-9 exactly mirrors our analysis of infinitely repeated games.To see why, suppose the firms adopt trigger strategies, whereby each agrees to charge a highprice provided the other has not charged a low price in any previous period. If a firm deviatesby charging a low price, the other firm will &#34;punish&#34; it by charging a low price until the gameends. For simplicity, let us assume the interest rate is zero so that the firms do not discountfuture profits.Given such trigger strategies, does firm A have an incentive to cheat by charging a lowprice? If A cheats by charging a low price when B charges a high price, A's profits are $50today but zero in all remaining periods of the game. This is because cheating today &#34;triggers&#34;firm B to charge a low price in all future periods, and the best A can do in these periods is toearn $0. Thus, if firm A cheats today, it earnsregardless of whether the game ends after one play, two plays, or whenever.If firm A does not cheat, it earns $10 today. In addition, there is a probability of 1 - 0 thatthe game will be played again, in which case the firm will earn another $10. There is also aprobability of (1 - 0)² that the game will not terminate after two plays, in which case A willearn yet another $10. Carrying out this reasoning for all possible dates at which the gameterminates, we see that firm A can expect to earnIICOPPA = 10 + (1 - 0)10 + (1 - 0)²10 + (1 - 0)³10 + ..100if it does not cheat. In this equation, 0 is the probability the game will terminate after one play.Notice that when 0 = 1, firm A is certain the game will end after one play; in this case, A'sprofits if it cooperates are $10. But if 0 < 1, the probability the game will end after one playis less than 1 (there is a chance they will play again), and the profits of cooperating are greaterthan $10.325

Accepted Answer

A game that is played for a finite number of rounds or duration and involves several players is…