A Practicum in Behavioral Economics 7 Some Classic Games of Iterated Dominance

CHAPTER . SOME CLASSIC GAMES OF DOMINANCE Before diving into the deep pool of behavioral game theory , we need some specific nomenclature about what constitutes a game and its solution , or what we have been calling its equilibrium . If you ve ever played a board or card game with your friends or family , then none of this terminology should surprise you . A game consists of a set of players , each with their own set of Precise rules govern the order in which players make their moves , the information they have available , and , ultimately , their I don know about you , but the card game poker comes immediately to mind . The keywords are players , strategies , rules , order , moves , information , and payoffs . We expect that Homo will attain what known as a Nash equilibrium , or perhaps a refinement of Nash equilibrium , depending upon the game being Simply put , a Nash equilibrium prevails when . The Nash equilibrium solution concept is attributed Nash , an American mathematician and the winner of the 1994 Nobel Memorial Prize in Economic Sciences . Nash published his pioneering BEHAVIORAL ECONOMICS

each player can no longer adjust his or her strategy to obtain added payoff . Thus , in a Nash equilibrium , all players have chosen respective strategies that are the best responses to each of the other players strategies . The Nash equilibrium is derived analytically and , thus , is highly predictable . We will see just how predictable the equilibrium is in a wide variety of games . Since this is the equilibrium obtained by Homo , we henceforth use the terminology Homo and analytical equilibrium . As we will learn , the equilibria typically obtained in games played by Homo sapiens expand upon the Nash equilibrium concept by adding in such aspects of the human experience as emotion , miscalculation , limited foresight , doubt about how informed the other players are , and of the same human quirks and idiosyncrasies we encountered in Section . The equilibria obtained in games played by Homo sapiens are typically derived more intuitively than analytically . Thus , the equilibria are generally unpredictable . Lets start with one of the most famous and basic of bargaining . ULTIMATUM BARGAINING Consider the following game presented in ( 2003 ) work in game theory in the early ( Nash , 1951 ) His struggles with mental illness and recovery are recounted in Sylvia 1998 biography A Beautiful Mind and later , the 2001 film of the same name . ust ( 2014 , pages ) provides a mathematical framework within which to assess the same outcomes of this game as We demonstrate here with more intuitive reasoning . 238 ARTHUR

Two Proposer and a bargain over 100 . The Proposer offers some portion , of the 100 to the Responder , leaving the Proposer with ( 100 ) If the Responder accepts the offer , then she gets and the Proposer gets ( If the Responder rejects the offer , both players get nothing . The analytical equilibrium for this game evolves according to the following logic By going first , the Proposer possesses all of the bargaining power . The Proposer , therefore , exploits the fact that the Responder will take whatever is offered . The amount offered by the Proposer is thus very close to zero . Surmising that this is indeed the Proposer best strategy , and also recognizing that he gets nothing if he rejects the offer , the Responder has no better strategy than to accept whatever the Responder offers , as meager as the offer is . The Proposer knows that this is the logic the Responder will use , and the Proposer knows that the Responder knows this , and so on . Hence , the analytical equilibrium is that the Proposer makes the meager offer ( in the limit , and the Responder accepts . Ouch . Before exploring what the behavioral game theory literature has to say about how Homo sapiens have actually played this game ( ie , what equilibria they have obtained ) it is informative to link this game to the nomenclature presented at the chapters outset . The players are a Proposer and Responder . The BEHAVIORAL ECONOMICS PRACTICUM 239

Proposer strategy is to choose an offer amount , that he thinks will ultimately be accepted and result in a desired payoff amount . The Responder strategy is to accept or reject the Proposer offer . The rules of the game , which govern which player moves when ( the order of moves ) and how the resulting payoffs are determined , are clearly spelled out . The Proposer moves first by making offer at and the Responder moves second , choosing to accept or reject the offer . After the Responder decision is made , the payoffs are distributed according to the following rule If the Responder accepts the Proposer offer , the payoffs are for the Responder and ( 100 ) for the Proposer if the Responder rejects the Proposer offer then the payoffs are zero for each . As the logic behind the determination of the analytical equilibrium makes clear , the information available to the Proposer and Responder has an important bearing on the games analytical equilibrium . Although this game has allocated all of the bargaining power to the Proposer , both the Proposer and the Responder are assumed to share complete and common information . Each player not only knows what payoffs he stands to gain via the games rule , but also what payoffs the other player stands to gain , and each player knows that the other player knows this , and so on . The fact that all players know the same things about the game is what is common about the information . The fact that no information is hidden from the players is what makes the information complete . The solution process for the Ultimatum Bargaining game analytical equilibrium follows what known as dominance due to ( the players making their moves sequentially ( or iteratively ) and ( the concomitant need for each player to think ahead about 240 ARTHUR

the other players subsequent move before choosing what to do presently . In this case , because each player best strategy is and unique , we say that it is Further , because the Proposer in this game initially considers what should happen in the last stage , where the Responder decides whether to accept or reject the offer made in the first stage , dominance is via backward The Proposer first figures out what should be the outcome of the games final stage and then works back from there to determine what she should do in each preceding stage all the way back to the first stage . Because the equilibrium is solvable via backward induction , we say that it is We will be seeing examples of backward induction and perfection repeatedly in this chapter , so get ready ! From an analytical , perspective , this is . An extreme form of Ultimatum Bargaining is known as the Dictator Game , whereby the Proposer makes an offer that the Responder must accept ( the Responder is not allowed to reject ) There is no iteration and no real role or advantage for having information . Yet , there is a dominant strategy for the Proposer , which results in an analytical equilibrium where the Proposer offers nothing to the Responder ( and the Responder accepts . Obviously , in this game , Responder is a euphemism for See et al . 1994 ) Hoffman et al . 1994 ) Hoffman et al . 1996 ) and and ( 2004 ) for experiments with the Dictator Game . Technically speaking , a perfect equilibrium is obtained when a Nash equilibrium has been reached at every of the original game , even if a particular has not been played . In the case of Ultimatum Bargaining , there are two one where the Responder either accepts or rejects the offer , and the other is the full game itself ( the full game is always considered a ) We will learn more about what defines a a bit later in the chapter . BEHAVIORAL ECONOMICS PRACTICUM 241

all interesting to know . But what about Homo sapiens ?

How have we actually played the Ultimatum Bargaining game ?

We have several different kinds of results . 2003 ) has compiled an exhaustive list of studies that have considered ultimatum bargaining with varying rules , payoff amounts , and multiple rounds , in different regions of the world with varied cultural , with men women , and more . He concludes that results from the different versions of the game are quite robust . Modal and median ultimatum offers are usually 40 of the total amount available to bargain over , and means are 30 . There are hardly any offers made by the Proposer in the outlying category of , and the category 51 . Offers of 40 are rarely rejected . Offers below 20 or so are rejected about half the time ( 2003 ) In other words , Homo sapiens do not generally converge to the games analytical equilibrium . It seems that are susceptible to emotions like guilt , fairness , or altruism , and Responders succumb to envy and fairness ( in this case , reciprocity ) Here is a taste of some of the findings Ironically , participants in cultures in Africa , the Amazon , New Guinea , Indonesia , and have been found to behave more like Homo than do participants in cultures in the US , Europe , and Asia ( and Roth , 1998 et , 2004 et , 2001 and 2002 , 2000 ) Repeated games with stranger matching and no provision of history of moves show a slight 242 ARTHUR

tendency for both offers and rejections to fall over time . Provision of history correlates with more pronounced reductions in offers and rejections ( Roth et , 1991 Bolton and , 1995 Knez and , 1995 and Roth , 1998 List and Cherry , 2000 ) Responders are not necessarily more likely to reject , say , out of 50 than out of 10 , and similarly 10 of 50 than 10 of 10 . In other words , the games stakes do not necessarily matter ( and Hogarth , 1999 Roth et , 1991 et , 1994 Hoffman et , 1996 and , 1995 Cameron , 1999 and Roth , 1998 ) Male do not necessarily offer more to attractive female Responders , but female have been found to offer more to attractive male Responders ( and , 1994 ) Young children are more , Homo and Responders , but then become more as they grow older ( Damon , 1980 and Saxon , 1998 et , 2000 ) Calling the game a exchange encourages . Describing the game as a common pool resource encourages generosity ( Hoffman et , 1994 and , 1997 ) Note that this is an example of a framing effect ! When know the exact amount of money to be divided , and Responders either know BEHAVIORAL ECONOMICS PRACTICUM 243

nothing at all or know the probability distribution of possible amounts , offer less ( Huck , 1999 and , 1993 and , 1993 and , 1995 , 1996 et , 1996 ) This is a consequence of incomplete However , when Responders know the alternative amounts that the Proposer could have offered , they tend to exhibit inequality aversion ( or , alternatively , a commitment to fairness ) and reject the Proposer offer ( Falk et , 2003 ) Creating a sense of entitlement by letting the winner of a contest ( played beforehand ) be the Proposer lowers offers ( Hoffman et , 1994 List and Cherry , 2000 ) This is known as an Entitlement Effect . 2017 ) eloquently sums up the main takeaway from these disparate findings Homo sapiens sense of reciprocity appears to with their economy structure , or if you like , the context within which the game is played . In addition to the varied described above , the structure of the Ultimatum Bargaining game has been modified as well . We consider two of these structurally adjusted versions of the Nash Demand Game and the Finite Game . NASH DEMAND GAME Consider the following game proposed by , et al . 1992 ) 244 ARTHUR

This game has three stages . Ultimately , at the third and stage , the two players individually state their If the two demands add to 10 or less , then they each get their individual demands otherwise , they each get nothing . In the stage , the players are each dealt four cards randomly from a deck with eight cards only four aces and four deuces ( twos ) Players are told that if all four aces are ultimately held by one player , then that player cards are worth 10 , in which case each player will have earned the right to state his demand in the third and stage . Otherwise , with any other of aces held by the two players , each player cards are worth nothing . In the second stage , the players trade their cards with each other . The analytical equilibrium for this game obtains evolves according to the following logic Since Homo know the composition of the deck , one player can tell from his own hand how many aces the other player , four minus his own number of aces . Thus , in the second stage , the players should always trade with each other such that the four aces end up being held by one of the players , as this gives them the right to state their demands in the third and final stage . Recognizing that the cards were randomly dealt to begin with , the players should each state a demand of in stage . What happens when Homo sapiens play the Nash BEHAVIORAL ECONOMICS PRACTICUM 245

Demand ( game instead ?

et al . provide an answer . The authors start by considering what happened when Player was dealt two aces in the first stage of their experiment . Of the 42 instances where this happened , 40 resulted in the player ultimately demanding half the pie of 10 in the final stage of the game . Thus , Player behaved as would be expected of Homo . However , when Player was dealt either one ace or three aces , she demanded half the pie only roughly half of the of 32 times when dealt one ace and 17 of 33 times when dealt three . The other half of the time , Player demanded a fraction roughly equal to the fraction of aces originally of 32 times when dealt one ace and 15 of 33 times when dealt three . As a result , there is a 22 ( 32 ) deviation from the analytical equilibrium ( of an even split in demands ) in cases where one and three aces have been dealt to Player . et al . postulate that the implicit information about how many aces each player originally held in the Game created focal points for this type of deviation . For example , when Player was dealt one ace and three deuces she was able to discern that Player held the other three aces and one deuce . In those instances , Player determined that because she contributed only one of four aces now held by Player she should demand less than half of the 10 . Similarly , when Player was dealt three aces she should demand more than half of the 10 . In a slight twist on the basic game , et al . 1998 ) had their subjects play the game with an outside This game is played according to the same rules as the basic game except that before the game begins , Player is randomly given a commonly known outside option worth , or . In other 246 ARTHUR

words , before the game begins , Player outside option ( which is equal to one of the five possible values stated in the previous sentence ) is announced to both players . Player can choose to take the option , in which case he gets that payment , and Player gets nothing . Or Player can turn down the option , and the game ensues . A Homo version of Player should ignore Player outside option since if Player turns down the option and opts to play the Game , then playing the game from that point forward is all that matters . Being a member of Homo , Player knows that this is indeed Player best strategy , and thus , if the game is ultimately played , Player will demand , which means that the most Player will be able to demand is also analytical equilibrium is therefore obtained . Hence , Player will take the outside option only if it is worth or . Otherwise , Player should turn down the option and play the game with Player . et al . found that Player do not behave like Homo . For instance , of Player opt out at the option value and only 60 opt out at . Further , the demands of those Player 23 who in at option values above match those ( focal ) values rather than the expected . Player behaviors deviate less from what we would expect of Homo . Their demands are relatively close to except in cases where Player option values exceed . Interestingly , Player demands decrease in accordance with the commonly known option values for Player , resulting in a total demand less than the 10 threshold . To the extent that Player expects Player demands to tilt toward the focal points of their option values , Player is actually making a rational choice in lowering her BEHAVIORAL ECONOMICS PRACTICUM 247

demands . And to the extent that Player with higher option values expects Player to lower her demand accordingly , then Player is likewise making a rational choice . The fact that relatively few Player make demands that leave Player with less than their option values is also rational to the extent that Player expect Player demands to tilt to their option values . So , while the behavior of the players in et ( 1998 ) game with an outside option does not adhere to those expected in an analytical equilibrium , to the extent that their behaviors are premised on Homo sapiens tendencies to tilt toward focal points in these types of games we can interpret the players as nevertheless making contextually rational choices . FINITE GAME Consider the following game presented in ( 2003 ) Two players bargain for two periods . In the period , Player offers a division of 200 to Player . If Player rejects Player offer , the pie of 200 shrinks to 50 and Player makes a counteroffer to Player . If Player rejects Player counteroffer , the game is over and neither player gets anything . Solving this game analytically requires the use of induction , which results in a perfect equilibrium ( SPE ) The logic goes like this Using backward induction , Player considers what 248 ARTHUR

Player will do in the second period when it is Player turn to make the counteroffer . Player I does not want the initial offer to be rejected since this will shrink the pie to 50 . So , Player offers at most 50 to Player ( 50 being the most Player could ever hope to get he rejects Player offer ) Player , therefore , keeps at least 150 and Player gets at most 50 . reports that in games played with Homo sapiens , Player tends to offer half of the pie in the first stage ( out of a sense of fairness or fear that the initial offer might otherwise be rejected by Player ) in which case Player ability to reject the initial offer is perceived as a credible threat by Player . However , with repeated play , Player I quickly learns to offer the SPE amount of 50 in the first stage . In other words , as Homo sapiens learn how to play the game , Player threat is no longer perceived as being all that credible . Incredible , huh ?

With learning , Homo sapiens attain the analytical equilibrium . CONTINENTAL DIVIDE GAME Consider the following game proposed by Van et al . 1997 ) Players ( more than two ) each pick a number from to 14 . The rows of the matrix below show each player payoff ( in dollars ) corresponding to the number she has chosen and the median choice made by the group as a whole . BEHAVIORAL ECONOMICS PRACTICUM 249

45 55 as 55 as as 405 435 as 92 55 51 777 as , As 54 so as 14 72 run As I 752 43 55 71 77 so 25 22 as an 52 so 69 71 as as 32 15 15 12 no 23 33 42 52 52 72 22 52 53 43 AI ' 39 ' as 13 23 an 54 75 75 ea 55 an 52 52 57 43 . 23 37 . 69 so no Ex 52 . 15 as 57 as as an 91 as as 11 15 Au 59 as nu nu as 94 mo 37 75 99 112 113 123 ) 456 437 412 57 94 mi 217 495 15 52 72 as 95 112 my ( Van Hu For example , if a player chooses and the median is , the player earns a healthy payoff of 71 . If the median is instead 12 , the player earns 14 ( she loses 14 ) Before discussing the logic behind the games analytical equilibrium , it is useful to point out Where backward induction and perfection factor into determining the equilibrium , if at all . It turns out that backward induction is actually a moot point in this game . This is because there is only a single players simultaneously choose their numbers , which then automatically determines the median number and attendant payouts . One might be tempted to say that there are 1471 , Where represents the number of different players . This is not correct . There are instead possible outcomes to what is only a single ( the game itself ) The logic behind the games analytical equilibrium goes like this First , note thatfor any median less than or equal to 250 ARTHUR

, a players best response is to choose the number . This is because , being a member of Homo , each player knows that every other player is both interested and thinking the same way . Thus , if each player chooses results in a median of will be in a given player to deviate and choose the number . But every player is equally and thinks the same way . Thus , a median of results . But at a median of , each player deviates to the number . And at a median , each player deviates to . Only at the choice of does this madness stop . We call this the low Nash equilibrium . Using the same logic , for any median greater than a players best response is to choose the number 12 . We call this the high Nash equilibrium . We expect this games analytical equilibrium to be the high Nash equilibrium . Van et al . played this game with 10 different groups of Homo sapiens , each group playing the game 10 times in a row . What they found were basins of attraction . For groups that start with a median of in the first period , the equilibrium converges to of , and ( there is an attraction toward the low Nash equilibrium ) For groups that begin with a median greater than in the first period , the basin of attraction leads toward of 12 and 13 ( the high Nash equilibrium ) Hence , while not all groups of Homo sapiens obtain the high Nash equilibrium , it seems that roughly half do . The outcome is what we call path dependent upon where the path . 2002 ) recounts a classic example of path Broken Window Theory . The theory is that a single broken window left unfixed in a neighborhood can lead to a spiraling process of social breakdown as those with criminal intent interpret the broken BEHAVIORAL ECONOMICS PRACTICUM 251

BEAUTY CONTEST Consider the following game presented in ( 2003 ) Each of players simultaneously chooses a number in the interval , 100 . The average for the group is calculated and then multiplied by a factor ( say , The player whose number is closest to this target ( in this case , 70 of the average ) wins 20 . Probably like you , the relationship between this game and a beauty contest escapes But the game , regardless of what we call it , provides a nice example of how dominance can be used to identify an analytical The logic for the analytical equilibrium is as follows Each player starts by thinking , Suppose the average is Given this , he chooses the number 35 ( 50 ) But he would not stop here ( he would begin ) window as a signal that the neighborhood is in decline and thus less able to protect itself . The broken window sets the neighborhood on the path toward a low Nash equilibrium . Keynes ( 1936 ) described the action of rational agents in an equity market using the analogy of a fictional newspaper contest where participants were instructed to choose the six most attractive faces from among a hundred photographs . Those who chose the most popular faces were then eligible for a prize . Note that this is a game where backward induction is again moot , and the only is the game itself . 252 ARTHUR

He realizes that everyone else is making the same calculation , so he will choose 25 instead ( 35 ) But wait . He would then choose 18 ( 25 ) But wait he would ultimately choose zero , which is this games analytical equilibrium . reports results for one Beauty Contest played by groups of Homo sapiens , where and there are low stakes of and high stakes of 28 . Irrespective of the stakes , Homo sapiens do not generally converge to the analytical equilibrium where each player chooses zero . But Homo sapiens do get close , especially when the stakes are higher . 2003 ) also reports that in most studies , players have used anywhere from zero to three levels of dominance , which , according to the logic for the analytical equilibrium , means that the numbers most frequently chosen are 50 , 35 , and a ways from zero . TRAVELER DILEMMA Consider the following game proposed by et al . 1999 ) Two players simultaneously state price claims , between 300 and 750 , for luggage lost by their airline company . The airline pays both players the minimum claim . The airline also adds a reward of 50 to the player who states the lower claim , and a penalty of 50 from the player who states the higher claim . BEHAVIORAL ECONOMICS PRACTICUM 253

Applying dominance , the logic for the analytical equilibrium goes like this Players should state claims that are one cent below what the other player is expected to state . In this way , a player helps boost the minimum claim ( and hence her payoff ) while earning the 50 reward . The result is a race to the bottom in which both players end up choosing the minimum claim of 300 , and thus , neither player wins the reward ( or , thankfully , suffers the penalty ) This is the games unique Nash equilibrium . Homo sapiens ?

et al . found convergence toward the analytical equilibrium with their subjects over 10 periods of play only for the higher penalty levels . In the later periods , average equilibrium claims were inversely related to the penalty levels ( the lower the penalty level , the higher the average claim ) Once again there is some evidence to suggest that Homo sapiens learn to converge toward ( not necessarily all the way to ) the analytical equilibrium , and the stakes of the game matter to some degree . ESCALATION GAME Spaniel ( 2011 ) explores the Escalation Game , depicted below as a decision . Decision trees in game theory are known as games depicted in extensive Drawing a decision tree is typically the most effective Way to depict a game . 254 ARTHUR

There are two players in this and Player . In the first stage , Player decides whether to Threaten Player or If Player accepts , the game ends with both Players and receiving payouts of each ( the number to the left of the comma denotes Player payout , and the number to the right denotes Player ) Player chooses to threaten Player in the first stage , then the game proceeds to the second stage where Player gets to choose whether to Escalate or If Player concedes , the game ends with Player receiving a payout of , and Player is required to make a payment to the experimenter of . If Player chooses to escalate in the second stage , then the game proceeds to the third and final stage where Player gets to choose War or to Give If Player chooses Give up , then the game ends with Player making a payment of to the experimenter and Player receiving a payout of . If Player instead chooses War , then the game ends with players required to pay the experimenter each . Note that depicting the game in extensive form makes it easy to identify the number of different . In this case there are four . BEHAVIORAL ECONOMICS PRACTICUM 255

10 . Can you guess the logic behind the games analytical equilibrium ?

Via backward induction , we start at the third and final stage and work our way back to the first stage . We see that Player will declare war ifthe game ever reaches its final stage since paying is a better outcome for Player than paying . Knowing this , Player choose to escalate in the penultimate stage since she will be required to pay as a consequence of war occurring in the final stage , which is a better outcome for Player than paying . But then knowing this , Player will choose to accept in the first period , which leads to a zero payout , which is , nevertheless , a better outcome than the payment of Player would be required to pay as a result of later going to war with Player . This is the games unique SPE . Note that in the case of international relations , this game captures the essence of mutual What drives mutual deterrence in the context of this game is that Player choosing to escalate in the penultimate stage acts as a credible threat to Player Whats the outcome when you and your classmates play this game ?

Hopefully , you choose mutual deterrence as opposed to going to war . ESCALATION GAME WITH INCOMPLETE INFORMATION Spaniel ( 2011 ) proposes a tweak to the Escalation Game Here a test to see how well you grasped the logic behind the game analytical equilibrium . What equilibrium would result if instead of having to pay by choosing to concede in the second stage , Player required payment was instead something less than ?

256 ARTHUR by endowing Player with less information than Player . We assume that Player does not know for certain whether Player is a weak or strong type . Player therefore assigns probability to Player being weak and ( to Player being strong . Nature has assigned Player his type , which Player alone is aware of with certainty . Player moves first . When he moves , Player knows both his type and the move made by Player in the first The decision tree for this game looks like this Nature ?

It In the initial stage ( which we will call Stage ) Nature determines whether Player is weak or strong . Player assumes Player will be weak with probability and strong with probability ( where . As 11 . This type of game is also known as a screening game , where the player moves first . BEHAVIORAL ECONOMICS PRACTICUM 257

indicated by the green hashed line , when Player moves in the first stage , she does not know for certain whether Player has been determined as weak or strong . If it turns out that Player was determined by Nature to be weak and Player concedes , the game ends with Player receiving a payoff of and Player receiving a payoff of . If , on the other hand , Player chooses to invade , then Player chooses between fight and concede in the second stage , resulting in payoffs of and and and , respectively for Players and . If , instead , it turns out that Player was determined by Nature to be strong and Player chooses to concede in the first stage , the game ends with Player again receiving a payoff of and Player receiving a payoff of . If Player chooses to invade , then again , Player chooses between fight and concede in the second stage resulting in payoffs of and and and , respectively to Players and . Whew ! Before working through the logic of the analytical equilibrium , notice that if and when the players reach the games second stage , Player will never choose to fight if he was determined by Nature to be weak ( remember that Player knows for certain whether he is weak or strong before play begins with Player ) This is because the payoff from conceding at that stage is , which is larger than the payoff of associated with choosing to fight . Similarly , if Player was determined by Nature to be strong , then he will never choose to concede if and when the players reach Stage ( Thus , the decision tree for this game can now be depicted as the following 258 ARTHUR

Now , how do we solve for the games analytical equilibrium ?

Here , Player applies backward induction to find whats known as a Perfect Bayesian Equilibrium ( As we already know , Player is the weak and Player I has chosen to invade , then Player should concede . If he is the strong type , then Player should fight . We also know that Player I recognizes that she gets a payoff of she concedes in the first round , 12 . This equilibrium is known as a Perfect Bayesian Equilibrium ( rather than an SPE because of the uncertainty that at least one of the players is forced to contend with . Similar to Nash , Thomas is considered a towering figure . He was an English statistician , philosopher , and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name Theorem . never published his theory notes were edited and published posthumously . BEHAVIORAL ECONOMICS PRACTICUM 259

regardless of Player type . If she instead chooses to invade in the first round , then Player expected payoff from invading is ( This is merely the weighted average of Player expected payoff when Player is weak and her expected payoff when Player is strong . Thus , invade is a better strategy than concede for Player when ) In other words , ifthe probability that Player assigns to Player being weak is greater than , Player should choose to invade in the first round . Otherwise , Player should concede and be done with it . Whats the outcome when you and your classmates play this more complicated Version of the Escalation Game ?

BURNING BRIDGES GAME This game shares starkly similar features with the Escalation Game , but there is no uncertainty ( thus , the analytical equilibrium is an SPE rather than a ) The SPE has much to say about the relationship between two tenacious competitors . Spaniel ( 2011 ) portrays the game as follows Suppose an island is located between two countries . Each country has a bridge to the island . 260 ARTHUR

Country decides to cross over its bridge to the island in an act of war . Country must then choose whether to burn the bridge behind it or not . The games structure is depicted by the following decision tree BEHAVIORAL ECONOMICS PRACTICUM 261 Recall that this game starts with Country already having crossed the bridge onto the island . Country choice in the first stage of the game is , therefore , whether or not to burn the bridge behind it . If Country burns the bridge , then Country must decide whether to cross its bridge and invade the island as well or to concede the island to Country . The resulting payoffs for the two countries are as shown . If , instead , Country chooses not to burn its bridge , then if Country also decides to invade the island , Country must then choose whether to stand and fight or retreat back over its bridge to safety . Otherwise , if Country decides to concede , then the result is the same as when Country decides to burn its bridge . 262 ARTHUR

The logic for the analytical equilibrium goes like this If Country chooses not to burn its bridge , Country will choose to invade the island knowing that Country will then choose to retreat ( since ) thus giving Country a , which is larger than the payoff it would have gotten had it instead chosen to concede . Thus , Country payoff from choosing not to burn its bridge is ultimately . If Country instead chooses to burn its bridge , Country will choose to concede ( since ) giving Country a payoff of . Thus , Country choosing to burn its bridge and Country responding by conceding the island to Country is this games SPE . There are historical examples of this game having been played between civilizations and countries and even individuals . For example , Collins ( 1989 ) recounts an incident in 711 AD when Muslim forces invaded the Iberian Peninsula , and commander Tariq bin ordered his ships to be burned , thus signaling to his troops that they had passed the point of no return . Harvey ( 1925 ) recounts a similar incident in ( formerly Burma ) In the Battle of , during the War in 1538 , the armies led by commander ( later known as ) faced the superior force of on the other side of a river . After crossing the river on a makeshift bridge , ordered the bridge to be destroyed . Similar to Muslim commander bin , took this action to spur his troops forward in battle and provide a clear signal that there would be no retreat . In both cases , the commanders were victorious . Have you ever burned your proverbial bridge in negotiations with an employer , a friend , or maybe even BEHAVIORAL ECONOMICS PRACTICUM 263

a family member ?

If the answer is yes , chances are you are not alone . Most Homo sapiens , if they live long enough , are eventually confronted with having to play a game like this . Typically , it requires a curious mixture of courage and desperation for a player ( in our case a Country ) to summon the will necessary to achieve the games analytical equilibrium . POLICE SEARCH Spaniel ( 2011 ) describes a game he once remembers having played himself . The title of the game says it all Suppose a police pulls over and asks to search his vehicle . Big Al can let the police officer search the vehicle ( which could be a quick or a thorough search , depending upon the police preferences ) or refuse and force the officer to call in the Canine Unit . Big Al preferences are Quick Search Canine Unit Thorough Search , while the police preferences are Quick Search Thorough Search Canine Unit . Without actually knowing Big preferences , the officer nevertheless claims that a Quick Search is more preferred for both of us than calling in the Canine Unit . Recall from Chapter that the symbol stands for strictly preferred Thus , We can say that based upon the information given above , Big Al strictly prefers a quick search as opposed to summoning the canine unit , 264 ARTHUR

and strictly prefers the canine unit as opposed to a thorough search . In contrast , the police officer strictly prefers the thorough search over a quick search , and strictly prefers the quick search over bringing in the canine unit . It helps if this game is depicted as a decision tree . Canine Unit Note that the hypothetical payoffs associated with each choice in the decision tree correspond with Big Al and the police officers respective preference rankings . To find this games analytical equilibrium , we need to assume common knowledge among Big Al and the officer ( each player knows both his own payoffs and those of the other player ) Common knowledge has been implicitly assumed for each of the games examined thus far except , of course , in the case of the Escalation Game with Incomplete Information . The logic for the analytical equilibrium is as follows If Big Al allows a search , the officer will choose to BEHAVIORAL ECONOMICS PRACTICUM 265

do a thorough search , implying Big Al payoff is and the officers is instead , Big Al does not allow the search , the Canine Unit is called in , resulting in payoffs to Big Al and the officer of and , respectively . the games SPE is Big Al not allowing a search , and the officer calling in the canine unit . In some sense , by holding his ground on not allowing the officer to search his car , Big Al is burning his bridge with the officer . The equilibrium outcome is driven by the fact that the officer can not credibly commit to conducting a quick search if Big Al were to allow the officer to conduct a search . Sadly , the resulting SPE for this game is inefficient since , as the officer originally pointed out , both Big Al and the officer prefer the quick search . If Big Al and the officer were Homo rather than Homo sapiens , they would mutually trust each other in this particular context , and the quick search would be conducted . Both the officer and Big Al would save Valuable time , and the canine unit would get more rest . DOMINANCE GAME Beard and ( 1994 ) propose the following game 266 ARTHUR

, 10 , Player chooses first , moving either left ( or right ( If she moves , the game ends with Players and receiving payouts of and , respectively . If , instead , Player moves , then Player chooses in the second stage whether to move left ( or right ( The payoffs to both players are then as given . The games SPE is determined via the following logic By backward induction , Player considers what Player will choose if the game proceeds to the second stage . Player will choose since . 75 . This results in Player , which is larger than the . she would obtain if she decides to move in the first stage . Thus , Player choosing and Player choosing is the games SPE ( denoted ( After playing this game with various groups of Homo sapiens , Beard and ( 1994 ) report that 66 of Player chose to move In the 34 of instances where Player 13 . These results are for Beard and ( 1994 ) baseline treatment . The authors considered several other treatments where the payoffs for the two players were modified . The results for most of these alternative BEHAVIORAL ECONOMICS PRACTICUM 267

moved , their choices were met with Player interested response of 83 of the time . Beard and calculated Player faith in Player rationality required to justify choosing in the first stage ( which they label a threshold probability ( as equaling . In other words , Player reported needing to believe that Player would choose in the second stage 97 of the time before they could justify choosing in the first stage . Since Player 23 chose only 83 of the time , the threshold was not quite met on average . DIRTY FACES ( 1953 ) invented this game whereby three ladies , A , and , in a railway carriage all have dirty faces and are all laughing . Because none of the ladies can see their own face to know for certain whether their face is dirty , they must infer from the laughter of the other two ladies whether their own face is dirty . The version of this game presented in ( 2003 ) involves only two players , but the notion of knowledge is nonetheless retained . Two players have independently and randomly drawn their types , either or , with probabilities of 80 and 20 , respectively . After observing the other player not their own treatments were qualitatively similar to those obtained in the baseline treatment . 268 ARTHUR

two players choose either Up or Down . Payoffs for each player are given in the matrix below . Type Thus , if a player chooses Up , he earns nothing . If a player chooses Down , he earns if he is type and loses 10 if he is type . When at least one player is type , both players are told , At least one player is type . Successive rounds of the game are played ( with each player retaining their original type ) until one of the players chooses Down . After each round , the players are told of the other player choice . The logic for the analytical equilibrium goes like this There are two cases to XO case ( one player is and the other is ) and ( both are ) We do not need to consider the OO case since , when this happens , each player will know immediately that he is an type ( how ?

and thus , neither of the players will ever choose the XO case , the player who is 14 . In actual experiments conducted by Weber ( 2001 ) with Homo sapiens , results for cases where both players drew the type are left unreported . BEHAVIORAL ECONOMICS PRACTICUM 269

can infer this fact ( how ?

He then moves Down . In case , both players know there is at least one type ( after the announcement is made that at least one of the players is ) and they know the other player is , but they still know nothing for certain about their own type . Each player , therefore , chooses Up in the first round and is then told of the other players choice . Player , for example , is told that Player chose Up . Player , therefore , that Player must have known Player was a type . Otherwise Player would have chosen Down . Thus , Player his own type from Player must be an . Hence , Player chooses Down in the second round . And therefore , for the case of XO , we expect the type player to choose Down in the first round of the game , while in the case , we expect both players to choose Down in the second round of play ( after the first announcement of player choices has been made by the experimenter ) Weber ( 2001 ) enlisted a small group of participants to play the Dirty Faces game . Recall that in Round , we expect the equilibrium to be ( Down , Up ) when the players are types and , respectively , and ( Up , Up ) when both players are type . In Round , played only by players who are both types , we expect the ( Down , Down ) equilibrium to result . The author found that in the XO case , player pairs behaved like Homo seven out of eight times across two different trials by choosing Down when they were type . In the trickier case , players are predicted to choose ( Up , Up ) in the first round , followed by ( Down , Down ) in the second round ( after each player is informed of the other players choice ) In Weber experiment , player pairs chose ( Up , Up ) in the first round 14 out of 18 times . But , only four of the 14 270 ARTHUR

player pairs who chose ( Up , Up ) in the first round chose ( Down , Down ) in the second round . The evidence for Homo sapiens is , therefore , mixed . They seem to mimic Homo in the XO case rather well , but not nearly as well in the case . TRUST GAME Consider the following game proposed by Berg et al . 1995 ) An Investor has 33 which she can keep or invest . Suppose she decides to invest and keep ( CE ) The investment of earns a return , at a rate of ( and becomes ( Another player , the Trustee , must now decide how to share the new amount ( with the Investor . Suppose the Trustee decides to keep and thus returns I ( to the Investor , resulting in a payoff of for the Trustee and ( ac for the Investor . Thus , is a measure of trust and I ( is a measure of trustworthiness . For our game , let 200 and . Despite the relatively complicated calculations involved in determining the returns to the Investor and Trustee for different possible investment and share values , the logic behind the games analytical BEHAVIORAL ECONOMICS PRACTICUM 271

equilibrium is a straightforward application of dominance practiced by the investor , in particular backward induction . Because the Investor anticipates that the Trustee will keep whatever investment is made , the Investor chooses to keep the entire 200 and thus invests nothing ! Consequently , in this games SPE , there is no trust displayed by the Investor and no opportunity for trustworthiness , or direct reciprocity , to be displayed by the Trustee . Not so with Homo sapiens . Berg et al . find that Investors invest roughly 50 on average ( 50 ) and Trustees repay roughly 95 of what was invested ( 95 ) which equals a negative return to trust and a correspondingly slight lack of trustworthiness ! In a modest tweak , et al . 2000 ) engage Asian and American subjects in a trust game where the Investor knows she will receive the return from a different ( Trustee rather than the original Trustee . The authors find that both trust and trustworthiness decrease relative to Berg et findings ( a sense of one would hope exists among not restore trust and trustworthiness ) et ( 2000 ) results are contained in the table below . 15 . The threshold for displaying trustworthiness in this game is 100 of the Investors investment returned by the Trustee . 272 ARTHUR

Pall Foursome Society Overall Percent Invested , Trust Level ) 75 49 49 54 Japanese 51 45 28 41 Mean 64 48 39 47 Percent Investment Returned . A Level Japanese 23 13 11 15 ( 41 25 IE 25 Mean 35 19 15 20 ( 2003 ) The different groups are similar ( when it comes to Investors , American and Chinese subjects behaved similarly to each other , and Japanese and Korean subjects behaved similarly as well ) As Trustees , American and Japanese subjects behaved similarly , and Chinese and Korean subjects behaved similarly . Pair represents a control treatment where the Investor receives a return from the same Trustee she invests with . The Foursome treatment refers to a version of the game where there are two Investors ( A and ) and two Trustees ( and ) Investor A originally invested with Trustee but is repaid by Trustee , and Investor originally invested with Trustee but is repaid by Trustee Investors A and know this cross repayment is occurring . The Society treatment refers to the case where Investors and Trustees are in separate rooms , and which Trustee repays which Investor is determined randomly . Investors A and , therefore , do not know which Trustee has been assigned to them ahead of time . The Overall column provides the average across these different treatments . We see that , on average , Investors chose to invest 64 of their initial amount with the Trustee in the control treatment ( which exceeds Berg et finding of 50 ) However , alternative pairings of Investors with Trustees BEHAVIORAL ECONOMICS PRACTICUM 273

lead to a reduction in the investment made by investors ( a decrease in trust ) Overall , trust is effectively displayed at a 47 level . Similarly , although in the control treatment 105 of the Investors initial investment is returned ( 35 105 ) the overall return on investment is only 60 ( 20 60 ) which represents a markedly lower level of trustworthiness than found by Berg et al ( 1995 ) Carter and Castillo ( 2011 ) conducted similar trust experiments with relatively poor and residents in rural and urban communities in South Africa . On average , Investors trusted their anonymous partners with 53 of their investable income , remarkably close to the percentages observed in the experiments performed by Berg et al . 1995 ) However , the amounts invested varied substantially depending upon which village the Investor was from . Similarly , on average , Trustees reciprocated in a trustworthy manner by returning 100 of the Investors investment . Carter and Castillo also found that trust and trustworthiness went located in villages with higher levels of trust also tended to exhibit higher levels of trustworthiness . Further , in urban communities , higher levels of trust and trustworthiness are correlated with higher levels of household expenditures ( a proxy for household ) while in rural communities this relationship is levels of trust and trustworthiness are associated with lower levels of household expenditure . One potential explanation for these latter results is that trust in a rural village is prone to moral hazard Oust , 2013 ) Moral hazard occurs when a persons actions are not fully observed by others , yet they can affect the 274 ARTHUR

16 . welfare of both that person and others . A trusting rural village might mean that residents generally assume that everyone else will perform their civic and economic duties and therefore do not need to be closely monitored . The marginal gain in household from trusting others a bit more is , therefore , relatively small . In a trusting village , residents will monitor each other more closely to learn whether others are actually performing their work , potentially leading to a larger marginal gain in household as a result of being able to trust others more . Alternatively , very few residents in an urban area are related to each other . This means that trusting others more might allow you to build a wider social network , potentially creating some substantial gains in household from being able to trust others 16 17 more . Stanley et al . 201 ) conducted trust games in the US with participants of different races and found that differences in trust and trustworthiness can be partially explained by differences in implicit attitudes toward race . Similar differences in trust and trustworthiness between races were discovered in earlier experiments conducted by et al . 2000 ) In their laboratory experiments , et al . 2009 ) find weak differences in trust and trustworthiness among individuals with different religious identities ( Hindu and Muslim ) and ( 1999 ) find no significant effect of gender on Investors level of trust . However , the authors find that women Trustees reciprocate significantly more of their wealth in trust games than men , both in the US and internationally . In a novel laboratory experiment , 2002 ) finds that trust game participants were more likely to trust others who look more like they do , which suggests that people over generations evolve toward promoting the and survival of others with whom they share a resemblance . Carter and Castillo ( 2004 ) conducted a trust experiment with survivors of Hurricane Mitch , which devastated rural communities in 1998 , with the goal of measuring the extent to which BEHAVIORAL ECONOMICS PRACTICUM 275

( TRUST GAME The trust game is best represented in the form of a decision tree a a wuss Player acts as the investor in the first stage by choosing whether to end the game immediately by not investing ( in which case she obtains a payoff of and Player as the Trustee in this nothing ) or by investing and continuing the game to the second stage . In the second stage , Player now acts as the Investor and decides whether to end the game by not investing ( in which case she obtains and Player as the Trustee in this ) or by investing and continuing the game to the third stage where , once again , Player acts as the Investor and Player the Trustee . As depicted , the game can be played up trust among community members helped spur recovery efforts . As the authors point out , while many communities received some of external aid , the absence of insurance contracts and thinness of capital markets meant that most households had to rely either on their own resources to muster an economic recovery , or on resources that they could broker through social relationships . Econometric analysis of the experimental data provided evidence of durable community norms , such as trust that is reinforced by social interactions . The analysis shows that trust played a strong , but uneven role in facilitating recovery from Hurricane Mitch , assisting most strongly a favored subset of households . While establishing the importance of norms such as trust , Carter and Castillo analysis warns against the presumption that all community members are equally by the social mechanism of trust in the face of recovery from a natural disaster . 276 ARTHUR

to 100 stages . And now we see how this game got its decision tree bears a striking resemblance to a centipede . The logic for the analytical equilibrium goes like this Via backward induction , Player ( as Investor ) ends the game in the final stage with no investment ( thus earning a payoff of 101 ) Knowing this will happen , Player I ( as Investor ) ends the game in the penultimate stage with no investment ( thus earning a payoff of 100 , which is larger than the 99 she would have earned as Trustee had the game instead ended in the final stage ) As the game unravels in mistrust all the way back to the first stage , Player I ( as Investor ) chooses to end the game in thefirst stage with no investment ( thus earning a payoff of only ) This game SPE is woefully inefficient . Too bad Homo are so interested . Had they been able to cooperate with each other they could have played to the final round , with Player earning 99 and Player earning 101 . 2003 ) survey of the literature suggests that Homo sapiens tend to reciprocate trust and trustworthiness for a few stages before one of the players ends the game . This may be a case of Player believing Player lacks common sense . Player thus plays Continue in the first stage , sending a signal that she trusts Player , who then also chooses Continue in the second stage . In cases where altruistic players are involved , the Honor payoffs in the final period may be interpreted as ( 101 , 102 ) and the players are and to advance all the way to the final stage . In a novel laboratory experiment , et al . 2001 ) led participants to believe that they were playing the Centipede game with a randomly paired partner . BEHAVIORAL ECONOMICS PRACTICUM 277

Before choosing a strategy , Player was given a photograph of the player ( Player ) to whom he was purportedly paired , and likewise for Player , who was given a photograph of purported Player . In reality , both players were playing against predetermined strategies programmed into a computer . Nevertheless , each player believed he was playing against the player in the picture . There were actually two pictures of each purported player . One of the photos depicted the player smiling , and the other depicted the player with a straight face . Participants were randomly assigned to see either a smiling or a partner . Overall , the authors found that participants were roughly 13 more likely to choose Continue at the first stage when their supposed partner was smiling in the photograph . Thus , there is some evidence to suggest that participants interpreted a smile on their partners face as signaling trustworthiness . Further , this smile effect was noticeably larger for male participants than for female participants . Male participants trusted a smiling partner roughly 20 more than a partner , while female participants trusted smiling partners only more . Additional experiments by the authors using other facial expressions also had an impact on the willingness of participants to trust each other . As the saying goes , whats in a smile ?

Perhaps the trust it inspires in those who are graced by one . To wrap up our exploration of the centipede game , consider and Palfrey ( 1992 ) version where , rather than the payoffs associated with successive stages of the game alternating between increases and decreases ( as depicted in the game above ) the payoffs increase at a constant rate for both players . Note that investigating 278 ARTHUR

possible effects associated with changes in the way payoffs evolve in the centipede game is reminiscent of investigations into the effects of changing the payoffs ( or stakes ) in the Ultimatum Bargaining and Beauty Contest games discussed earlier ( recall that changing the stakes in these two games generally had no impact on the games respective outcomes in experiments with Homo sapiens ) In their laboratory games , and Palfrey start with a total pot of divided into two smaller pots of and . Each time a player chooses to pass ( continue ) both pots of money are multiplied by two . The authors construct both a ( and a ( version of the game . and Palfrey also consider a version of the game in which all payoffs are quadrupled . The authors found that , as with the traditional Centipede game described above , the for these two versions of the game are for Player to take ( end ) the games in the first round . In each experimental session , and Palfrey use a total of twenty subjects , none of whom had previously played a centipede game . The subjects ( students from Pasadena Community College and the California Institute of Technology ) were divided into two groups at the beginning of the session , called the Red and Blue groups . In each game , the Red player was the first mover , and the Blue player was the second mover . Each subject then participated in ten games , one with each of the subjects in the other group . Subjects did not communicate with each other except through the choices they made during the game . Before each game , each subject was matched with another subject of the opposite color with whom they had not yet been previously BEHAVIORAL ECONOMICS PRACTICUM 279

matched . The paired subjects then played either the move or game . and Palfrey found that in only of the games , of the games , and 15 of the games did Player choose take in the first round . The subjects do not iteratively eliminate dominated strategies as they would if , like Homo , they played SPE In each of the sessions , the probability of a player choosing take increases as they get closer to the games last move . Thus , as subjects gain more experience with the game , their behavior mimics that of Homo . STUDY QUESTIONS Note Questions marked with a are adopted from Just ( 2013 ) those marked with a are adopted from Cartwright ( 2014 ) and those marked with a are adopted from Dixit and ( 1991 ) Recall the Ultimatum Bargaining game studied in this chapter . A Proposer makes an initial offer to a Responder of how to split 100 . If the Responder Instead , the players exhibit what on the surface appears to be altruistic behavior . However , as and Palfrey point out , if a selfish player believes that there is some likelihood that each of the other players may be altruistic , then it can pay the selfish player to mimic the behavior of an altruist in an attempt to develop a reputation for choosing to The authors surmise that the incentives to mimic are very powerful , in the sense that a very small belief that are in the subject pool can generate a lot of mimicking , even when the players face a very short time horizon . and Palfrey ultimately estimate that their sample consists of only . 280 ARTHUR

accepts the Proposer offer , the 100 is split accordingly . If the Responder rejects the Proposer offer then both receive . As we showed , the analytical equilibrium for this game is the Responder offering and the Responder accepting . How would the analytical equilibrium for this game change if the game instead adhered to the following rules In the first stage , the Proposer makes an offer to the Responder of how to split the 100 . In the second stage , the Responder can choose to either accept the offer as is , or agree to a of a fair coin . If the coin comes up Heads , then the game moves to the third stage . If the coin comes up Tails , the game ends with Proposer and Responder each receiving . In the third stage , the Proposer can decide to either split the 100 50 ( give 50 to the Responder and keep the remaining 50 ) or agree to a of a fair coin . If the coin comes up Heads , the Proposer keeps 75 and gives the Responder 25 . If the coin comes up tails , the Proposer instead gives the Responder 75 and keeps 25 . What is the analytical equilibrium for this version of the Ultimatum Bargaining game ?

Explain . Knowing what you know about basins of attraction , or path dependence , explain your strategy for playing repeated rounds of the Continental Divide game . What would the analytical equilibrium of the BEHAVIORAL ECONOMICS PRACTICUM 281 Beauty Contest game be if factor were instead set equal to a number greater than , say , Show how you arrive at your answer . Recall the Traveler Dilemma game where two travelers simultaneously submit claims to the airline for their lost luggage ranging between 300 and 750 . The airline pays both travelers the minimum claim , and then 50 from that amount for the player who submitted the higher of the two bids and adds 50 to that amount for the player who submitted the minimum of the two bids . In comparison with the analytical equilibrium for this game , explain why the airline should expect to pay out more in claims to two Homo travelers and less in claims to two Homo sapiens travelers if it changed the game accordingly The travelers get to choose one of two options . Option is the same as the original game , except now the on the range of claims is 250 instead of 300 . For Option , the airline a fair coin . If the coin comes up Heads , the traveler receives 750 if it comes up Tails , the traveler receives . Calculate the Perfect Bayesian Equilibrium ( for the Burning Bridges game if Country is uncertain as to whether Country has burned its bridge after it ( Country ) has occupied the island . Suppose you are a bank manager . You know that if depositors trust your bank , they will be willing 282 ARTHUR

to accept a lower interest rate on deposits . a ) Given what we know of how people develop trust , what might you do to enhance your depositors trust ?

What steps might you take to ensure that your loan officers can avoid potential pitfalls when it comes to originating loans to business owners and other customers ?

Suppose Alan and Emma are locked in a version of the Battle of The Sexes game depicted below ( you will be introduced to the classic simultaneous move version of this game in Chapter ) Alan chooses first , choosing to attend either the football game or ballet . Next , Emma chooses the football game or the ballet . The first value ( at each terminal node of the decision tree represents the payoff ( accruing to Alan , and the second value ( represents the payoff ( accruing to Emma ( all in dollars ) What is the most likely outcome of this game ?

Discuss how you have arrived at your answer . BEHAVIORAL ECONOMICS PRACTICUM 283 Emma Emma 33 20 . 10 . 10 ' i Now suppose that in her sequential game with Alan , Emma has an outside option that comes into play at the outset . She chooses to either go outwith her friends or throw her lot in with Alan for either the football game or ballet . What is the most likely outcome of the game now ?

Again , discuss how you have arrived at your answer . 284 ARTHUR . 10 . 11 . 20 . IO , 20 Choose a sequential game you learned about in this chapter . In what way might permitting communication between the players before the game begins affect the games outcome ?

Consumers who choose to purchase used vehicles from salespeople complain that the bargaining process resembles an ultimatum bargaining game . Why might this be the case ?

I In the game of roulette , betting is based on where a ball will land when a spinning wheel stops . In the games simplest form , there are numbers zero through 36 on the wheel . When the ball lands on zero , the players win nothing ( or , alternatively stated , the house wins ) The safest bet in roulette is simply to bet that the wheel will BEHAVIORAL ECONOMICS PRACTICUM 285

stop on an even or odd number , with numbers zero through 36 on the wheel the chances of Winning are , or a little less than 49 . This bet pays even money ( a bet returns , leaving the player with a total of ) A second possible bet would be that the wheel will stop spinning on a multiple of three ( the numbers , etc . for a , or slightly larger than a 32 chance of winning . This bet pays one ( a bet returns for a total of ) When players place their bets ahead of the spin of the wheel , they inevitably do so sequentially no rule says they must place their bets simultaneously with the spin of the wheel or with each other . Whenever a player bets wrong , places an bet on an odd number and the wheel stops on an even number , or , the player loses whatever amount of money was placed on the bet . Suppose that in this game the only possible bets are ( the bet on an even or odd number , or ( the bet on a multiple of three . Bonnie and Clyde are down to the last spin of the wheel . Whoever has amassed the most money ( in terms of the value of their chips ) at end of the final spin buys dinner . Bonnie has 700 worth of chips and Clyde has only 300 . All else equal , what is Clyde best bet ?

What is Bonnie best bet ?

Does either player have a mover advantage ?

12 . Suppose a new store called Newbies is considering entering a market that is currently 286 ARTHUR 13 . dominated by a store called Oldies ( Oldies is currently a monopoly in this market , earning ) It is common knowledge that if Newbies enters the market and Oldies accommodates ( does not wage a price war ) Newbies will earn and Oldies will also earn . If Oldies instead chooses to launch a price war , then Newbies will ultimately lose and Oldies will lose . Draw the decision tree for this game and determine its equilibrium . Suppose the state of Utah institutes a new statewide program called the Utah Brigades , which requires every high school senior to register for a year of public service to the state upon graduation . Worried that this new requirement may lead to mass civil unrest among the students , and unwilling to punish each student who refuses to register , the state announces it will go after evaders in alphabetical order by last name . a ) Explain why this approach could lead to full compliance with the registration . Would this approach still work if the state announced it will go after evaders by Social Security Number , in either ascending or descending order ?

Suppose two parents would like their adult children to communicate with them on a more regular basis . They announce a new quota that each respective child must meet in order to receive their portion of the parents inheritance one visit and two phone calls per week . Any child BEHAVIORAL ECONOMICS PRACTICUM 287

who does not meet the quota on any given week is disinherited , and the remaining children split the inheritance among themselves . Recognizing that their parents are very unlikely to disinherit all of them , the children get together and agree to cut back on their visits and phone calls , potentially down to zero . What change could the parents make to their will to ensure that the children meet their quota ?

Media Continental Divide Matrix Van et al . 1997 Science is licensed under a Rights Reserved license Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure ( Chapter ) Arthur is licensed under a BY Attribution ) license Burning Bridges Figure Arthur is licensed under a BY Attribution license Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure 13 ( Chapter ) Arthur is licensed under a BY Attribution license Figure 14 ( Chapter ) Arthur is licensed under a BY Attribution license 288 ARTHUR

Dirty Faces Figure Arthur is licensed under a BY Attribution license Table ( Chapter ) Colin Princeton University Press is licensed under a Rights Reserved license Figure 17 ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question ( Chapter ) Arthur is licensed under a BY Attribution license BEHAVIORAL ECONOMICS PRACTICUM 289