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Chapter . Game Theory Applications Repeated and Sequential Games Repeated Games A game that is played only once is called a game . Repeated games are games that are played over and over again . Repeated Game A game in which actions are taken and payoffs received over and over again . Many and relationships can be characterized as a repeated game . Strategies in a repeated game are often more complex than strategies in a game , as the players need to be concerned about the reactions and potential of other players . As such , the players in repeated games are likely to choose cooperative or strategies more often than in one shot games . Examples include concealed carry gun permits are you more likely to start a fight in a establishment , or one that allows concealed carry guns ?
Franchises such as were established to allow consumers to get a common product and consistent quality at locations new to them . This allows consumers to choose a product that they know will be the same , given the repeated game nature of the decision to purchase meals at . Sequential Games A sequential game is played in turns , or rounds like chess or checkers , where each player takes a turn . Sequential Game A game in which players move in turns , responding to each others actions and reactions . Product Choice Game One An example of a sequential game is the product choice game shown in Figure . Chapter . Game Theory Applications 185
WHEAT ( 10 , 10 ) OAT ( 10 , 10 ) Figure Product Choice Game One Cereal . Outcomes are in million . In this game , two cereal producers ( and General Mills ) decide whether to produce and sell cereal made from wheat or oats . If both firms select the same category , both firms lose five million , since they have flooded the market with too much cereal . However , the two firms split the two markets , with one firm producing wheat cereal and the other firm producing oat cereal , both firms earn ten million . In this situation , it helps both firms if they can decide which firm goes first , to signal to the other firm . It does not matter which firm produces wheat or oat cereal , as long as the two firms divide the two markets . This type of repeated game can be solved by one firm going first , or signaling to the other firm which product it will produce , and letting the other firm take the other market . Product Choice Game Two It might be that one of the two markets is more valuable than the other . This situation is shown in Figure . WHEAT ( 10 , 20 ) OAT ( 20 , 10 ) Figure Product Choice Game Two Cereal . Outcomes are in million . This cereal market game is very similar to the previous game , but in this case the oat cereal market is worth much more than the wheat cereal market . As in the Product Choice One game , if both firms select the same 186 Andrew The Economics of Food and Agricultural Markets
market , both lose five million . Similarly , if each firm chooses a different market , then both firms make positive economic profits . The difference between the two product choice games is that the earnings are asymmetrical in the Product Choice Two game ( Figure ) the firm that is in the oat cereal market earns 20 million , and the firm in the wheat cereal market earns 10 million . In this situation , both firms will want to choose OAT first . If is able to choose OAT first , then it is in General Mills best interest to select WHEAT . The player in this sequential game who goes first has a first player advantage , worth ten million . Each firm would be willing to pay up to ten million for the right to select first . In a repeated game , the market stabilizes with one firm producing oat cereal , and the other firm producing wheat cereal . There is no advantage for either firm to switch strategies , unless the firm can play OAT first , causing the other firm to move into wheat cereal . First Mover Advantage The first mover advantage is similar to the model of oligopoly , where the leader firm had an advantage over the follower firm . In many oligopoly situations , it pays to go first by entering a market before other firms . In many situations , it pays to determine the firms level of output first , before other firms in the industry can decide how much to produce . Game theory demonstrates how many firms determine their output levels in an oligopoly . First Mover Advantage Example Ethanol Ethanol provides a good example of the advantage . Consider an ethanol market that is a duopoly . To review the model , assume that there are two ethanol firms in the same market , and the inverse demand for ethanol is given by 120 , where is the price of ethanol in gallon , and is the quantity of ethanol in million gallons . The cost of producing ethanol is given by ( and total output is the sum of the two individual firm outputs . First , suppose that the two firms are identical , and they are . To solve this model , Firm One maximizes profits max ' max 111 ( price depends on total output max 120 max 211 120 max Chapter . Game Theory Applications 187
120 12 108 27 million gallons of ethanol This is Firm One reaction function . Assuming identical firms , by symmetry 27 The solution is found through substitution of one equation into the other . 27 ( 27 ) 27 18 million gallons of ethanol Due to symmetry from the assumption of identical firms 18 million gallons of ethanol , i gallons of ethanol 48 ethanol Profits for each firm are Iii ( Qi ) 48 ( 18 ) 12 ( 18 ) 48 12 ) 18 36 ( 18 ) 648 million This result shows that if each firm produces 18 million gallons of ethanol , each firm will earn 648 million in profits . This is shown in Figure , where several different possible output levels are shown as strategies for Firm A and Firm , together with payoffs . Next , suppose that the two firms are not identical , and that one firm is a leader and the other is the follower . By calculating the model solution , the possible outcomes of the game can be derived , as shown in Figure . In the model , assume that Firm One is the leader and Firm Two is the follower . In this case , Firm One solves for Firm Two reaction function max 112 max 112 ( price depends on total output max 112 120 188 Andrew The Economics of Food and Agricultural Markets
max 112 120 max 112 108 27 Next , Firm One , the leader , maximizes profits holding the followers output constant using the reaction function max ' max 111 ( price depends on total output max ' 120 max 111 120 max 111 120 ( 27 ) substitution of One reaction function max ' 120 54 max 65 max On 66 12 54 27 million gallons of ethanol This can be substituted back into Firm Two reaction function to solve for . 27 27 ( 27 ) 27 million gallons of ethanol 27 million gallons of ethanol 120 120 ( 120 81 39 ethanol ( 39 ) 27 27 ( 27 ) 729 million ( 39 12 ) 27 ( million These results are displayed in Figure . In a game , the Nash Equilibrium is ( 18 , 18 ) yielding payoffs of 648 million for each ethanol plant in the market . Each firm desires to select 18 million gallons , and have the other firm select million gallons , in which case profits would increase to 810 million . However , the rival firm will not unilaterally cut production to , since it would lose profits at the expense of the other firm . Chapter . Game Theory Applications 189
( GAL ) 729 , 729 ) 810 ) 729 18 ( 810 , 648 , 648 ) 324 , 486 ) 27 ( 729 , 486 , 324 ) Figure Advantage Ethanol . Outcomes are in million . In a sequential game , if Firm A goes first , it will select 27 million gallons of ethanol . In this case , Firm will choose to produce million gallons of ethanol , which is the solution . Firm A , as the first mover , has increased profits from 648 to 729 million by being able to go first . This is the first mover advantage . Threat Figure shows a sequential game between two grain seed dealers , a large international agribusiness , and a Local Grower . is the dominant firm , and chooses a pricing strategy first . If selects a HIGH price strategy , the Local Grower will select a LOW price , and both firms are profitable . In this case , the Local Grower has the low price , so makes more money than LOCAL GROWER HIGH PRICE LOW PRICE HIGH PRICE ( 100 , 80 ) 80 , 100 ) LOW PRICE ( 20 , 10 , 20 ) Figure Empty Threat Grain Seed Dealers . Outcomes are in million . Could threaten the Local Grower that it would set a LOW price , to try to cause the Local Grower to 190 Andrew The Economics of Food and Agricultural Markets
set a HIGH price , and increasing profits from 80 million to 100 million ?
could threaten to set a LOW price , but it is not believable , since would have very low payoffs in both outcomes . In this case , threat is an empty threat , since it is neither credible nor believable . Strike Suppose two big box stores are considering entering a small town market . If both and Target enter this market , both firms lose ten million , since the town is not large enough to support both firms . However , if one firm can enter the market first ( a strike ) it can gain the entire market and earn 20 million . The firm that goes first wins this game in a significant way This explains why has opened so many stores in a large number of small cities . ENTER DO ENTER ENTER ( 20 , DO ENTER ( 20 ) Figure Strike Big Box Outcomes are in million . and Credibility Figure shows a sequential game between beef producers and beef packers . In this game , the packer is the leader , and decides to produce and sell LOW or HIGH quality beef . Chapter . Game Theory Applications 191
LOW QUALITY HIGH QUALITY LOW QUALITY ( 20 , 50 ) 20 , PRODUCERS HIGH QUALITY ( 10 , 10 ) 40 , 20 ) Figure Commitment and Credibility One Beef Industry . Outcomes are in In . If the packers go first , they will select LOW , since they know that by doing so , the producers would also select LOW . This results in 50 million for the packers and 20 million for the producers . The producers would prefer the outcome ( HIGH , HIGH ) as their profits would increase from 20 to 40 million . In this situation , the beef producers can threaten the packers by committing to producing HIGH quality beef only . The packers will select LOW if they do not believe the threat , in the attempt to achieve the outcome ( LOW , LOW ) However , if the producers can commit to the HIGH quality strategy , and prove to the packers that they will definitely choose HIGH quality , the packers would choose HIGH also , and the producers would achieve 40 million . The producers could come up with a strategy of visibly and irreversibly reducing their own payoffs to prove to the packers that they are serious about HIGH quality , and cause the packers to choose HIGH also . This commitment , if credible , could change the outcome of the game , resulting in higher profits for the producers , at the expense of the packers . Such a credible commitment is shown in Figure , which replicates Figure with the exception of the LOW outcomes for the producers . If the beef producers sell off their low quality herd , and have no low quality cattle , they change the sequential game from the one shown in Figure to the one in Figure . LOW QUALITY HIGH QUALITY LOW QUALITY ( 50 ) PRODUCERS HIGH QUALITY ( 10 , 10 ) 40 , 20 ) 192 Andrew The Economics of Food and Agricultural Markets
Figure Commitment and Credibility Two Beef Industry . Outcomes are in In . If the packers are the leaders in Figure , they select the HIGH quality strategy . If they select LOW , the producers would choose HIGH , yielding 10 million for the packers . When the packers select HIGH , the packers earn 20 million . Therefore , a producer strategy of shutting down or destroying the low quality productive capacity results in the desired outcome for the producers ( HIGH , HIGH ) The strategy of taking an action that appears to put a firm at a disadvantage can provide the incentives to increase the payoffs of a sequential game . This strategy can be effective , but is risky . The producers need accurate knowledge of the payoffs of each strategy . The commitment and credibility game is related to barriers to entry in monopoly . A monopolist often has a strong incentive to keep other firms out of the market . The monopolist will engage in entry deterrence by making a credible threat of price warfare to deter entry of other firms . In many situations , a player who behaves irrationally and belligerently can keep rivals off balance , and change the outcome of a game . Political leaders who appear irrational may be using their unpredictability to achieve long run goals . A policy example of this type of strategy occurs during bargaining between politicians . If one issue is not going in a desired direction , a political group can bring in another issue to attempt to persuade the other party to compromise . The holdup game is another example of commitment and credibility . Often , once significant resources are committed to a project , the investor will ask for more resources . If the project is incomplete , the funder will often agree to pay more money to have the project completed . Large building projects are often subject to the holdup game . For example , if a contractor has been paid 20 million to build a campus building , and the project is only 50 percent complete , the contractor could halt construction , letting the completed building sit unfinished , and ask for 10 million more , due to cost This strategy is often effective , even if a contract is carefully and legally drawn up ahead of time . The contractor has the University right where they want it stuck with an unfinished building unless they increase the dollars to the project . The contractor is effectively saying , do it my way , or I Chapter . Game Theory Applications 193