A Practicum in Behavioral Economics 8 Some Classic Simultaneous-Move Games

CHAPTER . SOME CLASSIC GAMES DILEMMA This is undoubtedly one of the most simultaneous games Homo sapiens play ( for the most part unwittingly ) in their lives . Prisoner Dilemmas abound in our social interactions , in particular , governing how we manage natural resources collectively . You may have heard of the tragedy of the commons when it comes to managing fisheries , local and , or global climate change . It turns out that a Prisoner Dilemma is at the root of these types of resource management challenges . It is a dilemma that confronts us daily and drives individual making in a social setting . The game is presented below in its strategic form matrix containing the payoffs each of the two players will obtain from their respective choices when they move simultaneously as opposed to There is we have learned thus far , games where players move sequentially are generally depicted as decision trees ( or , in extensive form ) In 290 BEHAVIORAL ECONOMICS

common knowledge in this game in the sense that Player knows not only her payoffs listed in the matrix , but also Player . Player likewise knows not only his payoffs , but also Player . The payoff matrix for a Prisoner Dilemma game is depicted below Player Cooperate Deviate Cooperate I I Player Deviate In this game , both players simultaneously choose whether to Cooperate or Deviate . If both players choose Cooperate , then they both receive payoffs of each . If both instead choose Deviate , then they receive payoffs of only each . If Player chooses Cooperate but Player chooses Deviate , then Player receives a payoff of only while Player payoff jumps to . Likewise , if Player chooses Deviate when Player chooses Cooperate , then Player payoff jumps to and Player falls to . Because moves in this game are made simultaneously , the solution concept is not SPE or . Rather , it is either a pure strategy equilibrium or a mixed strategy contrast , games where players move simultaneously are generally depicted as payoff matrices ( or , in strategic form ) BEHAVIORAL ECONOMICS 291

It turns out that the Dilemma solves for a unique pure strategy equilibrium ( We will encounter simultaneous games that solve for mixed strategy equilibria ( a bit later in this chapter . The logic for this games analytical goes like this Both players consider their payoffs associated with Cooperate or Deviate given what the other player could decide to do and , in this way , devise their respective strategies . For instance , Player first considers her payoffs when Player chooses to Cooperate . Because , Player best strategy is to Deviate when Player chooses to Cooperate . Next , Player considers her payoffs when Player chooses to Deviate . Because , Player is best strategy is again to Deviate when Player chooses to Deviate . Because Player best strategy is to Deviate regardless of whether Player chooses to Cooperate or Deviate , we say that Player has a dominant strategy to choose Deviate no matter what Player decides to do ! Applying the same logic to Player decision process , we see that Player also has a dominant strategy to choose Deviate no matter what Player decides to do . Thus , the for this game is both players choosing to Deviate ( Deviate , Deviate ) Game theorists make a distinction between strictly dominant strategies and weakly dominant strategies . In the Dilemma , the strategies are strictly dominant because ( the payoff of choosing Deviate when the other player Deviates ( is greater than choosing Cooperate ( and ( the payoff of choosing Deviate when the other player chooses Cooperate ( is also greater than choosing Cooperate ( If either ( not both ) of these values were equal to each other ( the payoffs in the ( Deviate , Deviate ) cell were instead equal to each , or the payoffs in the ( Cooperate , Cooperate ) cell 292 ARTHUR

What a shame ! Both players choose to Deviate , and , as a result , they attain payoffs of only each . This equilibrium is woefully inefficient . Had the two Homo not been so and oh so rational , perhaps they could have agreed to Cooperate and earned each instead of just . Such is the essence ( and bane ) of the Prisoner Dilemma . Ironically ( or should I say , sadly ) when Homo sapiens play this game , they tend to attain the analytical , although cooperation has been found to occur in some experiments ( and , 2019 ) This should come as no surprise . As we now know , the Prisoner Dilemma ( and Homo sapiens proclivity for attaining the games ) is a contributing factor to historically intractable resource management problems in everyday life like air pollution , water scarcity , and climate FINITELY REPEATED DILEMMA Similar to the question of whether repeatedly playing the trust game for multiple stages could lead to greater trust and trustworthiness among an Investor and Trustee , the question arises as to whether repeated play of the Prisoner Dilemma can lead to more cooperation among two players in a . Unfortunately , applying backward were instead equal to each ) then the Deviate strategy would only be weakly dominant . Poundstone ( 1992 ) provides the onomatology of the title Prisoner Dilemma . As the title suggests , Dilemma was used to describe a fictional game where two suspects are apprehended , and the investigator wants both to individually confess to having participated in a crime . The investigator sets the prison sentences associated with confessing ( Deviate ) and maintaining innocence ( Cooperate ) such that the suspects dominant strategies are to confess . BEHAVIORAL ECONOMICS PRACTICUM 293

induction to the payoff matrix above for a finite number of periods suggests that the answer is To see this , suppose two players are in the final stage of the game . Given the payoff matrix above , Deviate , Deviate ) is the . Knowing this , in the penultimate stage , both players have no better options than to choose ( Deviate , Deviate ) again . Similar to the centipede game where mutual mistrust unfolds all the way back to the initial round , here , mutual deviation unfolds back to the first stage . Analytically speaking , cooperation does not emerge in a finitely repeated Prisoner Dilemma deviating is the dominant strategy for both players . Nevertheless , et al . 1982 ) and and Miller ( 1993 ) among others , find that Homo sapiens tend to cooperate more when they are uncertain of the other players tendency to cooperate . Furthermore , Spaniel ( 2011 ) demonstrates that when players adopt strategies such as grim trigger and in an infinitely repeated Prisoner Dilemma game , cooperation is more likely to occur . Grim trigger is a strategy where if your opponent Deviates at any stage , you Deviate forever starting in the next stage . Otherwise , you continuously Cooperate . is where you begin by choosing to Cooperate . In future stages , you then copy your opponents play from the previous PUBLIC GOOD GAME As in the Prisoner Dilemma , the dominant strategy in . In a more recent experiment , where players engage in what they interpret as an infinitely repeated Prisoner Dilemma game , Kim ( 2023 ) finds that higher discount factors ( as a result of or delays in when players are allowed to recoup their payoffs ) induce more cooperation among the players . 294 ARTHUR

a Public Good Game results in a among Homo , and often among Homo sapiens , that is inefficient when compared with What would otherwise be a cooperative outcome . In a simple version of this a linear public good is a group of players , each of whom receives an initial allocation of money , say 10 , and then is asked how much of that 10 she will voluntarily contribute to a group project of some kind . Each dollar that is donated by an individual player to the group project is multiplied by some factor a , and shared equally among all members of the group ( for the sake of a game played in a laboratory , the group project is simply a pot of money ) The fact that ensures that an individual players contribution to the pot of money is larger in value for the group as a whole than it is for that individual . For example , suppose there are four players ( including you ) and oz . If you contribute a dollar to the pot , then you give a dollar and receive only out of the pot in return ( players ) However , for every dollar contributed by another player , you receive out of the pot , free and clear . You can see how this game mimics the social dilemma Homo sapiens face when it comes to voluntarily financing a myriad of public goods ( or group projects ) such as public radio and public , environmental groups , and political campaigns , to name but a few . Each dollar contributed by someone else gives you additional benefit associated with a larger public good , for free . You get that same additional benefit from the public good when you are the one contributing but at the cost of your contribution . Since the overall return the group gets from your dollar contribution exceeds the dollar ( recall that in our case the BEHAVIORAL ECONOMICS PRACTICUM 295

overall return is per dollar contributed ) it is best ( socially efficient ) for each group member to contribute their full 10 allocation . The socially efficient equilibrium to this game earns each player a total of 20 ( 10 players ) players 20 ) In contrast , because each player only gets back for each dollar contributed , there is no individual incentive for a player to donate any amount of money ( or , to put it in economists terms , there is a strong incentive for each player to free ride on the generosity of the other players contributions ) Thus , the for this , the equilibrium we expect Homo players to obtain as a consequence of their , rational where each player free rides and contributes nothing . Grim , but true . One way to convince yourself that the for the Public Good Game is where each player contributes zero to the group project is to start at some arbitrary zero contribution level for each player and then show that each player has an incentive to reduce their contribution to zero . For example , suppose the starting point in our game is where each player contributes to the money pot . This means that each player would receive a total of 12 from the pot ( players ) players 12 ) Thus , each player takes home from the game a total of 12 16 . Now suppose one of the players decides instead to by dropping her contribution to . Each of the four players now receives from the money pot ( players ) player ) players ) The total pay for the three players is now , while the player takes home . Clearly , the player is better off by 296 ARTHUR

having dropped his contribution to , and the three players are each worse off . But then each of the would recognize that they too would have been better off by , just like the . So , they too have equal incentive to . Barring any type of made by each of the players , this process cascades to each player choosing to fully or , to use the Prisoner Dilemma lingo , to deviate from what is otherwise a fully cooperative equilibrium where each player contributes their total 10 to the money pot . Voila , we arrive at a where each player chooses to contribute zero to the money pot . Taking their cue from the likes of et al . 1982 ) and and Miller ( 1993 ) in testing a finitely repeated version of the Prisoner Dilemma Game , and ( 2000 ) explore whether a finitely repeated Public Good Game likewise mitigates deviation on the part of the players ( behavior ) The authors construct four treatment groups of student subjects . There is a stranger treatment , with and without punishment opportunities , and a partner treatment , with and without punishment ( punishment in this context is explained below ) In the partner treatment , 10 groups of four subjects each play the linear public good game for ten rounds without punishment and ten rounds with punishment , with group composition remaining unchanged across rounds ( hence , the title , partner treatment ) In contrast , in the stranger treatment , a total of 24 subjects are randomly partitioned into groups of four players in each of the twenty rounds ( 10 rounds without punishment and 10 rounds with punishment ) Group composition in the stranger treatment changes BEHAVIORAL ECONOMICS PRACTICUM 297

randomly from round to round . In both treatments , subjects anonymously interact with each other . Games played without punishment opportunities serve as a control for games played with punishment opportunities . In a game with punishment opportunities , each subject is provided the opportunity to punish any other player ( after any given round ) after being informed about each player contribution during that round . Punishment in this game takes the form of one player ( the punisher ) assigning punishment points to another player ( the punished ) For each punishment point assigned to a player , the players payoff from that round is reduced by 10 ( not to exceed 100 ) To mitigate the potential misuse of the punishment mechanism , face an increasing cost associated with assigning punishment points . The cost rises with the first two points assigned , and then rises at an increasing rate for points assigned beyond two . this is sounding a bit complicated . and results are depicted in the following two figures the first figure shows results for groups in the stranger treatment where the first 10 rounds are played without punishment , and the second 10 rounds are played with punishment . The expectation is that the availability of punishment opportunities would lead to an increase in the average player contribution to the money pot . This is depicted in the first figure both by the discrete jump in contribution level starting in round 11 , and the steady increase in this level over the remaining 10 rounds of the game . The downward trend in contribution levels over the first 10 rounds played indicates that the players learned that cooperation does not pay in a public good game without some form of punishment . 298 ARTHUR

20 18 16 14 I ' 12 without , I I Periods ( and 2000 ) The second figure shows similar results for the treatment groups . What is notable in comparison with the results for the groups is that ( the downward trend in the initial 10 rounds becomes noticeably steeper from round seven onward , and ( the initial jump up in average contribution level starting in round 11 ( when punishment opportunities become available ) is markedly larger , leading to higher average contributions levels thereafter . These results demonstrate what is commonly known as effects associated with a players history of contributions over time . Among groups , effects are enabled , while among treatment groups , they are not . BEHAVIORAL ECONOMICS PRACTICUM 299

a 5533 . Avenge I A 10 12 15 19 20 ( and 2000 ) It appears that punishment works with Homo sapiens in repeated play of a Public Goods Game , similar to how punishment works with Homo sapiens in repeated play of . a Dilemma Game . In an intriguing comparison of the effectiveness of punishment in finitely repeated public good games , et al . 2010 ) find a surprising result regarding the punishment of average contributors . For example , in the Australian city of Melbourne and the European cities of , and Copenhagen , there is little punishment for contributors , while in the Saudi Arabian city of , the Greek city of Athens , and the Belarusian city of , contributors were punished at the same levels as contributors . The authors call this latter form of retribution social punishment , attributed to their observation of a strong correlation between punishment received in one period and that doled out in the next . For example , if Sam punished Sally in period , Sally then punished Sam in period as revenge . Because punishment is ultimately associated with less cooperation among players , this study findings serve as a 300 ARTHUR

In addition to enabling punishment opportunities as a coordinating mechanism to reverse the grim , inefficient , equilibrium among Homo sapiens in a finitely repeated Public Good Game without punishment , Rondeau et al . 1999 ) and Rose et al . 2002 ) propose a promising mechanism for games , called the Mechanism . As Rose et al . explain , a provision point mechanism solicits contributions for a public good by specifying a provision point , or threshold , and a guarantee if total contributions do not meet the threshold . Extended benefits are provided when total contributions exceed the threshold . The authors report that the provision point mechanism has led to increased contribution levels ( and thus adequate funding for public goods ) in their laboratory and field STAG HUNT As its name suggests , this game tests the extent to which hunters can coordinate their efforts to bring down big game . 2004 ) explains the games Stag Hunt is a coordination game in which two hunters go out on a hunt together . Each can cautionary tale for those who espouse punishment as a universal remedy to the problem . With their laboratory experiments , Chan et al . 2002 ) sought to answer the question of whether involuntary transfers for the provision of a public good , such as taxation , crowd out voluntary transfers ( private donations ) that a emotion among subjects . The authors do not find evidence of complete crowding out in general , but suggest that crowding out increases as the rate of taxation increases . Sufficiently large rates of taxation offset the benefits of giving . BEHAVIORAL ECONOMICS PRACTICUM 301

individually choose to hunt a stag or a rabbit . If one of the hunters hunts a stag , she must have the cooperation of the other hunter to succeed . Thus , like in the Prisoner Dilemma , choosing to cooperate is other hunter can indicate he wants to cooperate but , in the end , take the less risky choice and go after a rabbit instead ( remember , like in the Prisoner Dilemma , the players decisions are made simultaneously in this game ) Alone , a hunter can successfully catch a rabbit , but a rabbit is worth less than a stag . We see why this game can be taken as a useful analogy for social cooperation , such as international agreements on climate change . An individual alone may wish to cooperate ( reduce his environmental footprint ) but he deems the risk that no one else will choose to cooperate as being too high to justify the change in behavior that his cooperation entails . Here is the games payoff matrix Player We use the same logic to determine this games 302 ARTHUR

analytical as we did to determine the Prisoner Dilemma Player first decides what to do if Player chooses to hunt stag . Because Player payoff in this case from hunting stag ( exceeds his hunting rabbit ( Player will choose to hunt stag when Player hunts stag . Next , we see that when Player chooses to hunt rabbit , Player will also choose to hunt rabbit since , in this case , the payoff from hunting rabbit ( exceeds the payoff from hunting stag ( Using the same approach to determine what Player best strategy is , we see that she will also choose to hunt stag when Player hunts stag and will hunt rabbit when Player hunts rabbit . Hence , neither player has a dominant strategy in this game , and as a consequence , there are actually two . One is where both players hunt stag , the other is where both hunt rabbit . We can not say for sure which of the two equilibria will be obtained . Clearly , the ( stag , stag ) equilibrium is preferable ( also known as dominant ) But this equilibrium requires that credible assurances be made by each player . In contrast , the ( rabbit , rabbit ) equilibrium is risk dominant in the sense that by choosing to hunt rabbit both players avoid the risk of having gone for stag alone . We would expect this equilibrium to occur when neither player is capable of making a credible assurance to hunt stag . Also note that even though this game does not was a Italian engineer , sociologist , political scientist , philosopher , and economist . He made several important contributions to economics , particularly concerning the study of income distribution and analysis of individuals choices . He is considered one of the fathers of Welfare economics . BEHAVIORAL ECONOMICS PRACTICUM 303

permit the use of backward induction by the players ( as a result of the game consisting of simultaneous rather than sequential moves ) each player inherently uses forward induction in predicting what the other player will choose to do . et al . 2019 ) recently conducted an experiment where a random sample of individuals playing a series of Stag Hunt games are forced to make their choices about whether to hunt stag or rabbit under time constraints , while another sample of players has no time limits to decide . The authors find that individuals under time pressure are more likely to play stag than individuals not under a time constraint . Specifically , when under time constraints , approximately 63 of players choose to hunt stag as opposed to 52 when no time limits are imposed . GAME Consider the following payoff matrix ( where , as with the Prisoner Dilemma and the Stag Hunt , each player payoffs are common knowledge ) 304 ARTHUR

Player Left Right Up Player Down The reason why this matrix depicts a game is because the payoffs to Players and sum to zero in each cell . Any time a player wins , the other player loses . You may have heard someone say , my gain is your loss or the other way around , or perhaps you ve said something like this to someone yourself . When this happens , the two individuals are ( unwittingly or not ) acknowledging that they are participating in a game . with the Prisoner Dilemma and Stag Hunt games , environmental and resource economists recognize that the global fight against climate change exhibits features of a game . All else equal , whenever one country invests in its carbon emissions , all other countries gain by not having to expend funds themselves to get the same amount of reduced carbon emissions . In terms of who bears the opportunity cost of the investment , this is one country loss ( or at least attenuated gain ) and every other country gain ( since units of carbon reduced anywhere on the planet reduce the atmospheric stock of carbon that is responsible for rising ground temperatures and other meteorological changes occurring across the planet ) BEHAVIORAL ECONOMICS PRACTICUM 305

Using the same logic as we used in the Prisoner Dilemma and Stag Hunt games to determine the players best strategy , we find that there is no for this game . Player best strategy is to choose Up when Player chooses Left , and Down when Player chooses Right . On the contrary , Player best strategy is to choose Right when Player chooses Up , and Left when Player chooses Down . No emerges . What is Homo to do ?

It turns out that the analytical equilibrium solution concept for games such as this is whats known as a equilibrium ( where players choose probabilistic mixtures in which no single strategy is played all the time . For instance , if I always choose a particular strategy , and you anticipate that strategy , then you will win . I should , therefore , behave more unpredictably . is a sensible strategy for me to follow when a little genuine unpredictability will deter the other player from making a choice that leads to a suboptimal outcome for me . The equilibrium involves unpredictable mixing on both the players parts . To facilitate the role randomization plays in determining an , We amend the game payoff matrix to account for each player probabilistic moves . 306 ARTHUR

Player 29 UP Player Down Now we suppose that Player chooses Up with probability ( and thus , Down with probability ( and Player chooses Left with probability ( and Right with probability ( It turns out this games occurs when ( Player chooses such that Player is indifferent between choosing Left or Right ( Player expected payoff from choosing Left equals her expected payoff from choosing Right ) and ( Player simultaneously chooses such that Player is indifferent between choosing Up or Down ( Player expected payoff from choosing Up equals his expected payoff from choosing Down ) In particular Player I chooses such that ( 149 ) Player chooses such that ( I ( BEHAVIORAL ECONOMICS PRACTICUM 307 ?

Thus , Player ( as a member of Homo ) chooses Up half the time , and Player ( also a member of Homo ) chooses Left half the time . Probably the best way each player can be true to their respective strategies is to a fair coin ( for Player , it might be Heads go Up , tails I go Down , and similarly for Player ) Lets see why the equality ( 11 ) holds when Player expected payoff from choosing Left equals her expected payoff from choosing Right ( you then be able to see why ( 161 ) I ( holds when Player expected payoffs from choosing Up and Down are equated ) When Player chooses Left and Player chooses Up , Player payoff is . The probability of Player choosing Up is , hence Player expected payoff from choosing Left , conditional on Player choosing Up , is ( Similarly , when Player chooses Left and Player chooses Down , Player payoff is . The probability of Player choosing Down is ( hence Player expected payoff from choosing Left , conditional on Player choosing Down , is ( We then sum these two values ( plus ( to attain Player ( unconditional ) expected payoff from choosing Left . The same process is followed to determine Player expected payoff from choosing Right . Setting these two expected payoffs equal solves for . To be clear , the for the game does not always result in To see this , calculate the equilibrium Values for ) and II when the payoff matrix is something like ( Up , Left ) Up , Right ) Down , Left ) and ( Down , Right ) 308 ARTHUR

The proof for why an is determined by each player their choice such that the expected payoffs for the other player are equated across that players choices is simply proved by contradiction . If , for example , Player his choices such that Player expected payoff is larger when she chooses Left than Right ( because Player can see that ) Player will always choose Left . Because he more often chooses Down when , Player payoff is lower than it otherwise would be if he instead chose The same logic holds when Player chooses , in which case Player always chooses Right . Because he more often chooses Up when , Player payoff is again lower than it otherwise would be if he instead chose . This is because Player payoff is again guaranteed to be when . Thus , the best Player can do is set . The proof is the same for Player . Hence , we have proved why an is is the optimal outcome for this each player their choice such that the expected payoffs for the other player are equated across that players choices . Whew ! 2003 ) informs us that in studies with choice games ( games where the two players play the game two times consecutively ) Homo sapiens tend to use the same strategy after a win but switch strategies after a loss . This , heuristic is a coarse version of whats known as reinforcement In games ( players play four times consecutively ) Homo sapiens strategies are remarkably 10 . This is because Player payoff is guaranteed to be when , BEHAVIORAL ECONOMICS PRACTICUM 309

close to predictions by the fourth time they play the game . When I was teaching in , this was the one game I was called into service to play myself ( we had an odd number of students that day ) By then , I had gotten to know my students quite well individually . I was designated Player , and my student ( I will call her Sally ) was designated Player . We were playing a game due to time constraints imposed on the course . I had observed Sally over the previous weeks and concluded that she was unlikely to just a proverbial coin in deciding whether to choose Left or Right . She was and always sat to my right in the classroom . I guessed she would choose Right . So , I chose Down . My guess was , luckily for me , proven correct by Sally . Afterward , when I explained my strategy to the students , I emphasized that a player need not actually randomize his moves as long as his opponent can not guess what he will do . An can therefore be an equilibrium in beliefs , beliefs about the likely frequency with which an opponent will choose different strategies . But I reminded the students that we had only played a game . With repeated play , chances are Sally would have begun to randomize her choices , to the point that a coin would become my best strategy as well . We would slowly but surely evolve from Homo sapiens to Homo . STAG HUNT ( REPRISE ) In our first assessment of the Stag Hunt game , we learned that two exist , with no way of definitively determining which of the two are most likely to occur . 310 ARTHUR

As a result , we are compelled to determine the games , as this is as close as we can get to identifying a unique analytical equilibrium . The games payoff matrix is reproduced here , this time accounting for the players probabilistic moves Player ( Stag Rabbit Stag Player ( Rabbit Using the same procedure as shown for determining the analytical for the game , the value of for Player is determined as ) and the value Player is determined as ( Each hunter might as well flip a fair coin in deciding whether to hunt stag or rabbit . It turns out that the for this game results in expected payoffs for each player that are larger than the certain payoffs obtained when both hunt rabbit ( which , you recall , is one of the games ) but lower than the payoffs obtained when both hunt stag ( the games other BEHAVIORAL ECONOMICS PRACTICUM 311

) To see this , we can calculate Player expected payoff for the as ( Dissecting this equation , the first term is Player payoff in the ( Stag , Stag ) cell of the matrix multiplied by the probability that both players will choose to hunt stag ( Similarly , the second payoff in the ( Stag , Rabbit ) cell multiplied by the probability that Player chooses to hunt stag and Player chooses to hunt equal to ( This is another way of saying that , when faced with the Stag Hunt , a coin essentially leads to a bit less than splitting the difference for each player from jointly hunting stag and jointly hunting rabbit ( technically speaking , splitting the difference would result in payoffs of each ) Cooper et al . 1990 and 1994 ) used the following payoff matrix as the baseline ( or what they call the Control Game ( for their Stag Hunt game experiment 312 ARTHUR

Player Stag Rabbit Stag , 800 Player Rabbit 800 , 800 , 800 A clear majority of their Homo sapiens pairs who participated in this game obtained the inefficient ( Rabbit , Rabbit ) equilibrium . The authors also had different of subjects play what they called ( the , where Player could opt out and award both players 900 instead of playing the game ( note that 900 800 ) the , where Player could opt out and award both players 700 instead of playing the game ( note that 700 800 ) where one of the two players is allowed to engage in cheap talk with the other player , presumably to nudge the other player into committing to hunt stag and ( where both players are allowed to engage in cheap talk in an effort to nudge each other into committing to hunt stag . Cooper et al . found that 97 of player pairs chose the ( Rabbit , Rabbit ) in the treatment . A large number of Player Is also took the outside option in the treatment . In cases where Player I refused the outside BEHAVIORAL ECONOMICS PRACTICUM 313

option , more than a of player pairs ( 77 ) obtained the efficient ( Stag , Stag ) equilibrium . In the treatment , the majority of player pairs reverted to the inefficient ( Rabbit , Rabbit ) equilibrium . Lastly , with cheap talk between players the efficient ( Stag , Stag ) equilibrium jumps from in the treatment to 53 . Unexpectedly , the jump is even greater for cheap talk ( up to 91 ) These results are encouraging for Homo sapiens because by simply allowing players to communicate with each other ( presumably to hunt stag ) Homo sapiens are , for the most part , capable of attaining the efficient outcome where both players hunt stag together . In cases where an outside option is available for one of the players , as long as that option payoff is larger than the mutual payoffs associated with the ( Rabbit , Rabbit ) yet lower than the mutual payoffs associated with the ( Stag , Stag ) the player with the outside option conjectures that the other player will choose to hunt stag , who is likely to end up confirming that conjecture . BATTLE OF THE SEXES While it is unclear who actually named this game , there is little debate about the genesis of the titles popularization , which occurred on Mothers Day in 1973 at the dawn of the womens liberation Tennis stars Bobby Riggs and Margaret Court faced off in a challenge match , which then Riggs , a tennis champion from the late and who was notoriously dismissive of women talents on the tennis court , resoundingly won . Later that year , Riggs The following interpretation is taken from ' 2020 ) 314 ARTHUR

challenged the tennis star Billie Jean King to a challenge match , which King won handily . Although the game we have in mind here is far from being a sports match between the sexes , it does capture the of challenges that sometimes bedevil couples coordination decisions . The payoff matrix for this game is presented below . Like the Stag Hunt , the games analytical equilibrium consists of two ( can you identify them ?

Therefore , to determine the games unique , we acknowledge Spouse and Spouse probabilistic strategies upfront . Spouse ( Ballet Martial Arts ?

Ballet , Spouse ( Martial Arts I I Given what you have just learned about solving for in the Stag Hunt and games , you can show that for this game , and . Further , the expected payoff in the for each spouse is . Interestingly , these expected payoffs are lower than the least preferable payoff in either of the games two , where the spouses either agree to watch the martial arts performance or attend the ballet . Recall that in the Stag BEHAVIORAL ECONOMICS PRACTICUM 315

Game the expected payoffs for each player split the difference of payoffs from the two . In Cooper et ( 1989 ) experiments with Homo sapiens , subjects mismatched 59 percent of the time , which is actually an improvement over in the games analytical . occurs in the when one player chooses Ballet and the other Martial Arts . This occurs ( 67 of the time . The authors also found that when one player ( say , Spouse in our game ) is given an outside option ( which , if Spouse takes , pays him and his spouse some value , such that ) and the husband rejects the option , the analytical equilibrium would entail Spouse surmising that Spouse will then choose Martial Arts . Thus , Spouse should also choose Martial Arts . In their experiment , Cooper et al . 1989 ) found that only 20 of Spouse chose the outside option . Of the 80 of Spouse who rejected the option , 90 obtained their preferred Arts , here we come ! With communication , the players coordinated their choices 96 of the time ! However , with simultaneous communication , they coordinated only 42 of the time ! What happened ?

Recall that , in the Stag Hunt game , communication enhanced coordination . Here , in the Battle of the Sexes , it has the opposite effect . Lastly , Cooper et al . 1989 ) found that when one of the players is known to have chosen ahead of time , but the other player is not informed about what the other player chose , the mismatch rate between the players decreased by roughly half ( relative to the baseline game with no communication or outside options ) 316 ARTHUR

PENALTY KICK There are few team sports where an individual player is put in as precarious a position as a goalie defending a penalty kick in the game of soccer ( or football if you are from anywhere else in the world except the US , Canada , Australia , New Zealand , japan , Ireland , and South Africa ) Homo sapiens rarely look more vulnerable than when put in a position of having to defend a relatively wide and high net from a ball kicked from only 12 yards away . Spaniel ( 2011 ) provides a nice analogy in the context of a payoff matrix where we are forgiven for taking the liberty of depicting the analytical equilibrium as an upfront . Do the payoff combinations for the striker and goalie in each cell of the matrix ring a bell ?

The bell should be ringing game . Goalie I ( Striker ( For this game , we assume a superhuman ( as opposed BEHAVIORAL ECONOMICS PRACTICUM 317 to a mere Homo ) goalie . If the striker kicks Left ( and the goalie guesses correctly and dives , the goalie makes the save for certain . Similarly , if the striker kicks Right ( and the goalie correctly dives , the goalie again makes the save for certain . The striker , however , is fallible . If she kicks and the goalie dives , she scores for certain . But if she kicks and the goalie dives , she only scores with probability av . Using the method we previously developed to solve for an in the Stag Hunt , and Battle of The . Further , for those of you who know Sexes games , verify that , in the equilibrium , and . calculus , you can use these equations to solve for the respective first partial derivatives of and with respect 12 12 . as aw Herein lies the closest thing to understanding the likely choices that are made by the goalie and The first partial derivatives together inform us that , in an , as the strikers probability of scoring goes up when she kicks ( embodied by an increase in ) the probability of the striker kicking actually goes down . Analytically speaking , it is as if the striker uses one degree of knowledge to determine the kick direction . The more the goalie believes the striker has a higher probability of scoring when she kicks , the more likely the goalie will dive Thus , it makes more sense for the striker to kick The next time you watch a game with lots of penalty kicks , you will be able to test this equilibrium concept in your own experiment with Homo sapiens . Note that solving for the likelihood that a goal will actually be scored on any given penalty kick is , to put it mildly , anyone guess . 318 ARTHUR

GAME No , this is not a game played among . The game is named after Harold , a mathematical statistician and theoretical economist who pioneered the field of spatial , or urban , economics . The game is depicted below Suppose there are two vendors ( and ) on a long stretch of beach selling the same types of fruit juices . The vendors simultaneously choose where to set up their carts each day . are symmetrically distributed along the beach . The buy their from the nearest The line graph below distinguishes the furthest south location on the beach at and the furthest north location at . lfyou are one of the two vendors where would you decide to locate your cart on the beach ?

The analytical equilibrium for this game is a . The logic behind its solution goes like this locates at she guarantees herself at least half of the total amount of business on any given if doesn also locate at . Indeed , locating at maximizes her . To see this , start 121 at and at . Hold 122 at I and move toward . Note that commands the most exposure to at location BEHAVIORAL ECONOMICS PRACTICUM 319

. Given that chooses location , it is in best interest to also locate at ( using the same logic we used to determine that 121 would locate at ) Thus , is the for this game . It should be no surprise that this result is also known as the Median Voter Theory throughout a typical campaign , candidates for public office tend to gravitate toward the middle of the political spectrum , toward the median Collins and ( 2000 ) studied how , and Games were played among Homo sapiens . The authors found that , in games , Homo sapiens strategies are remarkably close to the analytical prediction of ( In games , the analytical occurs at locations A and . Homo sapiens cluster at these locations , but also somewhat in the middle as well . In games , the occurs where each player his locational choice uniformly over the interval of locations between A and inclusive , denoted 14 , 34 . Thus , location intervals , and ( are avoided . In games , relatively smaller percentages of players locate outside the interval of 14 , 34 and larger percentages of players locate within the to go Homo sapiens ! Homo would have located strictly within the interval , with each player his choice uniformly over the interval . IS MORE INFORMATION ALWAYS BETTER ?

The model of Homo answers yes , in most cases . The more information the information can aid a consumer ARTHUR making . But wait . Spaniel ( 2011 ) offers a game where the answer is surprisingly no , more information is not always better , even for Homo . Consider the following game Player chooses whether to play or not . If Player chooses not to play , both Players and get 100 . Simultaneously to Player choice , Player chooses between Heads ( and Tails ( on a coin , or chooses not to gamble on the coin . If Player had chosen to play and Player had chosen not to gamble on the coin , both players receive 200 . If instead Player chooses to gamble , the coin is . If Player has called the outcome of the correctly , she wins 300 and Player loses 300 if Player has called the outcome of the incorrectly . Therefore , the game looks like this Player I Quit Play , 200 , 200 Quit Note that the ( payoffs in the cells ( Play , and ( Play , represent the players respective expected payoffs since whether the players win or lose 300 depends upon Player luck in correctly predicting the outcome BEHAVIORAL ECONOMICS PRACTICUM 321

of the coin . The analytical equilibrium for this game goes as follows Because Player weakly dominant strategy is to Quit , the for this game is for Player to choose Play and Player to choose Quit ( Play , Quit ) Note that this equilibrium is efficient ! Most importantly for our purposes , Player gets 200 in this equilibrium . Now we consider a slight tweak to this game where Player has private information about the outcome of the coin before he decides whether to play or quit . In other words , Player now knows whether the coin has come up Heads or Tails beforehand . Thus , Player is assured of winning 300 if Player has no prior knowledge of the outcome of the coin to Play . The key question is whether Player will now choose Play with positive probability ( If not , then Player has been harmed by having private information about the outcome of the coin wins only 100 , instead of 200 . Recall that , in the previous game , Player chose whether to play or quit before the coin was . She was , therefore , uninformed about the outcome of the coin before deciding whether to play the game . Now , suppose Player chooses whether to play or quit after the coin is , and that the outcome of the coin is Player private information . The players are now involved in known as a Bayesian Nash Game , where there are effectively two types of Player type ( with a 322 ARTHUR

probability of and a type ( with a probability of ( The game now looks Ike I 51 ' The analytical equilibrium goes as follows Unfortunately for Player , Player will set . If Player sets , then Player will simply play the correct side of the ifit was and ifit was ensuring a win of 300 when Player Plays and winning 100 when Player Quits . But then , Player wins 300 when she Plays , implying that Player will never choose to Play ( she will never choose ) Thus , the analytical this game is ( Quit , Quit ) with each player winning 100 . Since 100 is less than What Player won when he did not possess private information about the outcome of the coin ( which was 200 ) more information in this context is not better for either player . I dont know about you , but there are plenty of instances where having less information to sift through before making a decision eases my mind and actually makes me feel happier . For example , I tend to assemble appliances , furniture , or equipment with much less angst when the instructions are concise and to the point , preferably accompanied by clear pictures or diagrams for me to follow . Lengthy BEHAVIORAL ECONOMICS PRACTICUM 323

written descriptions often prey upon my insecurities and impatience . MARKET ENTRY Consider the following game proposed by 2003 ) Each of 20 players decides privately and anonymously whether to enter or stay out of the market . For each period , a different market carrying capacity , denoted by odd integer ( is announced , after which the players make their entry decisions into the market . Each player payoff ( is calculated as if the player stays out of the market ( if the player enters the market where 100 , and represents the total number of players who enter the market . Note that at the time of their decisions , each player knows the values of , and 20 , but obviously not Example If , and in equilibrium , a player who decided not to enter the market that period receives 100 , and a player who entered the market receives 100 ( 106 . As ( 2003 ) shows , the analytical equilibrium for this game is rather complicated . 324 ARTHUR

At the market level , there are two Nash Equilibria per value of For example , let . then each of the 20 players earns 100 . If one of the seven entrants instead chooses to stay out of the market , then she will not increase her payoff above 100 ( she now earns 100 as a ) If instead , one of the games 13 entrants decides to enter the market , this new entrant will decrease her payoff from 100 to 98 . Thus , no player has a profitable deviation from this equilibrium . For example , let . then each of the eight entrants earns 102 , and each of the 12 earns 100 . of the eight entrants instead chooses to stay out of the market , she will decrease her payoff by ( from 102 to 100 ) of the games 12 decides to enter the market , then this new entrants payoff remains 100 ( since now ) Thus , again no player has a profitable the equilibrium . At the individual player level , there is a unique . Letting represent the probability of any given player deciding to enter the market for a given , and letting represent the total number of players , a given player expected payoff from entering is calculated using the expression for the binomial distribution , we we ) For background on the binomial distribution see , BEHAVIORAL ECONOMICS PRACTICUM 325

13 . Thus , an individual player reaches the when , mar He is ( when the player is indifferent between entering the market ( expected payoff represented by the side of the equality ) and staying out of the market ( certain payoff represented by the side of the equality ) which can be 19 Thus , in equilibrium , In their laboratory experiments , et al . 1995 ) found that Homo sapiens mimic the analytical equilibrium rather closely . In a baseline setting ( Experiment ) 20 subjects were provided with no feedback between 60 rounds of series of six blocks , each block based upon a randomly chosen value of ( 11 , 13 , 12 , 17 , 19 ) were played by each subject 10 times each . In the first block , one of the 20 subjects chose to enter the market ( in ) when , four subjects entered ( when , seven entered ( when , and so on . et al . ultimately find a relatively close correspondence between the mean values of and their associated values of across the six blocks . Recall that in the analytical equilibrium in ( and should be roughly equal ) This close correspondence is supported by relatively large ( close to one ) correlation coefficients reported for each block ( with a coefficient of for the 60 rounds in total ) This means that on average in Thus , each subject was presented with each Value of ( six different times over the course of the experiment . 326 ARTHUR

and moved in roughly the same direction over the 60 a relatively large value , so was , and when relatively small , so was ( where implies perfect linear correlation between the two values ) Second , variability in the market entry decision across participants was largest for intermediate levels of In a second experiment ( Experiment ) subjects were provided with feedback at the end of each round regarding the equilibrium value of , as well as their respective payoffs for each round and cumulative payoffs up to the round . Subjects were also encouraged to write down notes concerning their decisions and outcomes . Results for this experiment were even closer to those predicted for Homo in the analytical equilibrium . As ( 2003 ) points out , how firms coordinate their entry decisions into different markets is important for an economy . If there is too little entry , prices are too high and consumers suffer if there is too much entry , some firms lose money and waste resources , particularly if fixed costs are . Public announcements of planned entry could , in principle , coordinate the right amount of entry , but announcements may not be credible because firms that may choose to enter have an incentive to announce that they surely will do so in order to ward off competition . Government planning may help reduce this perverse incentive , but governments are nevertheless vulnerable to regulatory capture by prospective entrants seeking to limit competition . Evidence from the real world often suggests that too many firms choose to enter markets in general , particularly in newly forming markets . As we learned in Section , the phenomenon of BEHAVIORAL ECONOMICS PRACTICUM 327

excessive entry could a Confirmation Bias on the part of potential entrants , leading to overconfidence in their abilities to obtain positive WEAKEST LINK Consider the following game presented in ( 2003 ) Players ( more than two ) pick numbers from to . Payoffs increase with the minimum of all the respective numbers chosen , and decrease with the deviation of their own choice from the minimum . The payoffs ( in dollars ) for our game are shown in the table below 14 . and ( 1999 ) designed an experiment to detect the extent to which overconfidence in ones skills relative to What he perceives are the skills of potential competitors induces excess entry into a market and subsequent business failure . The authors uncover a phenomenon they termed reference group neglect , Where excess entry is much larger when subjects participated in the experiment knowing that payoffs would depend upon skill level . These selected subjects neglected the fact that they are competing with a reference group of subjects who all think they are as skilled . 328 ARTHUR

Lowest Choice in Group ( 2003 ) For example , if a player chooses and the minimum chosen by another player in the group is , then that player payoff is . You may recognize this game as being a generalization of the Stag Hunt and reminiscent of the Continental Divide . As such , there is an efficient equilibrium where each player chooses the number , and an inefficient , dominant equilibrium where each player plays it safe by choosing the number . Choices of and act as potential basins of attraction for , while choices of and are potential basins of attraction for . Van et al . 1990 ) played this game with a total of 107 Homo sapiens into groups consisting of between 14 and 16 subjects for 10 rounds . The authors found that relatively large numbers of the subjects choose higher numbers in the first players chose the number seven , 10 players chose the number six , 34 chose BEHAVIORAL ECONOMICS PRACTICUM 329

five , and 17 chose . However , by the tenth and final round , 77 players chose the number one and 17 chose the number . These results are obviously discouraging . Recall that the efficient equilibrium is where there is no weak link among the player chooses the number seven , and no players choose the number one . Interestingly , results for 24 Homo sapiens who played the game for only seven rounds were more encouraging . In the first round only players chose number seven , zero chose number , and four chose number . By the seventh and final round , 21 had chosen the number seven . Perhaps these results are unsurprising given what we ve already learned about Homo sapiens behavior in the Finitely Repeated Trust ( Centipede ) Game . In that game , we learned that in repeated play , trust and trustworthiness between two players can be achieved , at least temporarily . Random the of trust and trustworthiness that can arise through repeated play . And , generally speaking , it is more difficult to build trust and trustworthiness among a larger group of players . As for applications of the Weakest Link Game , 2003 ) points out that in the airline business , for example , a weakest link game is played every time workers prepare an airplane for departure . The plane can not depart on time until it has been fueled , catered , checked for safety , passengers have boarded , and so on . For carriers , which may use a single aircraft on multiple flights daily , each departure is also a link in the chain of multiple flights , which creates another weakest link game among different ground This suggests that efficient equilibria have been obtained in an airline company weakest link games if all of a passenger connecting 330 ARTHUR

WEAKEST LINK WITH LOCAL INTERACTION ( TWO VERSIONS ) We now consider two different versions of the Weakest Link Game presented in ( 2003 ) In the first version , players are provided with some information at the end of each round about the choices made by the other players . In the second version , players play separate games simultaneously with their neighbors , thus permitting the experimenter to test whether spatiality might affect the games equilibrium outcome among Homo sapiens . Here is version one To begin , players individually choose the letter or in separate groups of three players each over 20 rounds . Payoffs ( in dollars ) are shown in the table below . Each player learns the two choices made by the other two players at the end of each round ( but not which player made with choice ) are on time . Whether or not any given passenger makes it to the airport and his connecting on time is his set of weakest link games . BEHAVIORAL ECONOMICS PRACTICUM 331

Two Other Players One , Both One Both 80 60 60 Row Player 10 10 90 And here is version two In this version of the game , players face the same payoff matrix Two other Players One , Both one Both Row Player 10 10 But each group of eight players is arranged in a circle , and each subject plays with his two other nearest neighbors 332 ARTHUR

For example , in this diagram Player plays with Players and directly , while Player simultaneously plays with Players and and Player simultaneously plays with Player and . Note that the , inefficient equilibrium in versions occurs when each player decides to play it safe by choosing option . The efficient equilibrium occurs when all three players in a group choose . In this way , the game mimics the Weakest Link Game introduced earlier . In keeping with Van et ( 1990 ) original findings , We would therefore expect that , because the game here is being played in relatively small groups ( three players per group ) Homo sapiens playing these games would again mimic Homo and obtain the efficient equilibrium , especially in version one of the game . Version two presents a complication since , BEHAVIORAL ECONOMICS PRACTICUM 333

although the groups are small , each player now plays with multiple sets of different players simultaneously , either directly or indirectly . The players do so directly ( Player simultaneously plays with Players and , on the one hand , and Players and on the other ) players do so indirectly ( Player plays with Players and , who in turn play simultaneously with Players and and Players and , respectively ) In playing this game with different groups of Homo sapiens , et al . 2002 ) found that in standard ( groups , players initially play about of the , in their experiments seven out of eight groups of three players coordinated on the efficient ( equilibrium . No surprise . However , in the ( circular ) groups , players started by playing only half of the time , but then play of fell steadily to almost none by the round of the game . Specifically , players responded to their neighbors by playing 64 of the time when one other neighbor had chosen . The authors likened this result to the spread of a disease through a population by close this case , the spread of fear . The incidence of playing spreads from neighbor to neighbor , eventually infecting the entire group . A merger where one or more disparate groups of individuals merge into a single group is an extreme form of local interaction , one that is continually played out through history on a ( beginning with families merging into clans who merge into ethnic groups , to villages and to and sometimes empires ) On a scale , mergers occur when one company acquires another and becomes its 334 ARTHUR

parent Mergers and acquisitions among businesses are a common feature of market would say they pose a threat to competition , while others believe they can provide efficiency gains via economies of scale . and Knez ( 1994 ) wondered how the Weakest Link game might inform us about the efficacy of mergers occurring between two small groups into a single larger group . The effect noted earlier suggests that , all else equal , larger ( merged ) groups would be less likely to converge to the efficient equilibrium where all group members individually select the number seven . This is exactly what and Knez found . Interestingly , the provision of public information ( to each group about the other group with which it later merged ) appears to have worsened the outcomes ( increased the inefficiency ) in paired small groups of three players each . and Knez conclude that the results for merged groups in the context of a Weakest Link game are not encouraging . However , not to be outdone by these results , the authors added another treatment to the mix ( in addition to the provision of public information about the performance of the other groups in the previous round ) The additional treatment was a public announcement to all groups of a bonus if everyone picked the number ( if each merged group chose the efficient outcome ) Thankfully , the announcement nudged the merged groups immediately in round , from 90 choosing numbers one or two to 90 choosing number . and Knez Homo sapiens apparently need a little added incentive merger to attain an efficient equilibrium in their link choices . BEHAVIORAL ECONOMICS PRACTICUM 335

CONCLUDING REMARKS Section has introduced you to the burgeoning field of behavioral game theory , a field that investigates how Homo sapiens play several of the classic games devised by game theorists to depict expected outcomes ( analytical equilibria ) when people interact in a variety of social situations . In particular , behavioral game theory identifies how Homo sapiens deviate from a games analytical equilibrium and devises tweaks to the game in an effort to gain insight into what might be causing the deviation . We began our exploration of the field in Chapter by studying the classic bargaining Bargaining , Nash Demand , and Offers . The distinguishing features of these games are ( they are played sequentially ( one player makes a choice first and then the other player chooses ) the analytical equilibrium is premised upon each player thinking ahead about the other players subsequent move before choosing what to do presently ( each player thinks iteratively ) and ( the process of thinking iteratively requires that each player first considers what she should choose to do in the games final stage , and then work backward to what to do in the games initial stage . The resulting equilibrium is known as perfect solved via backward induction and of an dominance in the players respective strategies . As complicated as all of this sounds , we learned that Homo sapiens often achieve this type of equilibrium in a variety of . Likewise , we studied a variety of games with perfect equilibria involving sequential 336 ARTHUR

moves by the players , each in their own way demonstrating the concept of dominance . Recall the Escalation , Burning Bridges , Police Search , Dirty Faces , and Trust Games . In the Dirty Faces game , the trajectory toward the games analytical equilibrium depends upon an initial , random draw of cards . One draw quickly leads players to the equilibrium the other draw necessitates more intensive knowledge on the part of the players . As expected , Homo sapiens do a better job of attaining the equilibrium when less knowledge is required of them . In the Trust game , Homo sapiens players demonstrate excessive trust and insufficient trustworthiness . Next , we considered a series of games in this chapter where players are again expected to use iterative thinking but this time in where decisions are made simultaneously , not sequentially . As a result , players are expected to use forward induction in predicting how their opponents will play rather than backward induction . In some of these games , the analytical equilibria exist in pure strategies , where the payoffs are structured in such a way as to make strategies dominant over others and the corresponding equilibria inefficient . This was the case with the famous Prisoner Dilemma where , unfortunately , Homo sapiens tend to mimic Homo and obtain the inefficient equilibrium . However , when Homo sapiens are allowed to play the game repeatedly , cooperation among players ensues . Other simultaneous games do not exhibit dominant strategies , leading instead to equilibria where players choose their strategies randomly but in accordance with some rule , such as a fair coin . BEHAVIORAL ECONOMICS PRACTICUM 337

The Stag Hunt , Game , and Battle of The Sexes are examples of games with equilibria . Empirical research suggests that Homo sapiens are capable of attaining a games equilibrium , but only with the aid of additional incentives such as or communication ( known as cheap talk ) among players , or the availability of outside options for one of the players that the other player is also aware of . Similar results were found for other games such as the Market Entry and Weakest Link Games . In the Market Entry Game , Homo sapiens responded to both and cumulative feedback by more closely mimicking the analytical equilibrium . In Weakest Link Games , smaller player groups achieved the analytical equilibrium more often . But when players played two games game with a different set of analytical equilibrium was obtained less often . Therefore , it appears that Homo sapiens tend to perform more like Homo in social settings ( in games played with multiple players ) than in individual settings ( the laboratory experiments presented in Section ) particularly when appropriate incentives are provided in the social setting . Homo sapiens tend to learn in both settings through repeated play and with appropriate incentives . For example , with appropriate incentives , if the payoffs are set too low or too high , players with access to options are more likely to choose strategies that deviate from the analytical equilibrium . This has been demonstrated in Stag Hunt games . As we learned from experiments with Ultimatum Bargaining and the Beauty Contest , the level of a games stakes itself 338 ARTHUR

does not generally the probability that players will attain an analytical equilibrium . As we proceed to Section , recall that the experiments and games we have studied in Sections have traditionally been conducted with relatively small samples of university students . As such , the of their results to wider populations is justifiably drawn into question . In Section , we explore empirical studies based upon larger and more diverse ( more representative ) samples of individuals , the results from which are , by design , more generalizable to wider populations . 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 ) Consider the Prisoner Dilemma game depicted below in which two prisoners are accused of a crime . Both are isolated in the prison . Without a confession , there is not enough evidence to convict either . A prisoner who confesses will be looked upon with lenience . If one prisoner confesses and the other does not , the prisoner not confessing will be imprisoned for a much longer time than if she had confessed . The payoffs for the prisoners are depicted in the matrix below ( Prisoner payoffs are to the right of the comma in each cell , and Prisoner are to the left of the comma in each cell ) a ) If a selfish prisoner plays BEHAVIORAL ECONOMICS PRACTICUM 339

this prisoner dilemma against an opponent she believes to be altruistic , what will her strategy be ?

Now suppose the prisoners dilemma is played three times in sequence by the same two prisoners . How might a belief that the other prisoner is altruistic affect the play of the selfish prisoner ?

Is this different from your answer to part ( a ) What has changed ?

Prisoner Do ) 111 ) Inn Prisoner ) 30 , In discussing strategies for a finitely repeated Prisoner Dilemma game , it was mentioned that may be a strategy that induces cooperation among players in an infinitely repeated version of the game . Recall that a player adopts a strategy when he cooperates in the first round of the game and then , in each successive period , mimics the choice ( deviate or cooperate ) made by his opponent in the previous period . Thus , if Player plays and Player deviates in the first round , then Player 340 ARTHUR

commits to deviating in the second round , and so forth . If Player believes that Player is also playing , and the games payoff matrix is the same as that originally presented in this chapter ( repeated below for reference ) is it still in Player interest to play ?

Explain . Player CU ( Cooperate , I Player ) I . i Consider the game below , played between Man and Emma . Alan payoffs are represented by the first dollar value in each cell of the matrix , and Emma payoffs are represented by the second dollar value . Identify each of this games equilibria . Which equilibrium would you prefer and Emma to reach together ?

What incentive might you give Emma and Alan to ensure that they would reach the preferred equilibrium ?

BEHAVIORAL ECONOMICS PRACTICUM 341 Emma Low Effort High Effort Low Effort Alan High Effort , 13 , 13 . Choose a simultaneous game you learned about in this chapter . In What way would permitting communication between the players before the game begins affect the games outcome ?

In the discussion of the Public Good game , it was mentioned that a Mechanism has been found in experiments to increase contributions to public goods . Explain why this is the case . i Describe the similarities and differences between the Weakest Link and Threshold Public Good games , where the threshold in the public good game is a mechanism . Is reaching an international agreement on the control of climate change similar to a Stag Hunt game ?

Explain . 342 ARTHUR . 10 . and Donald face the game presented below . Calculate the games strategy equilibrium . or Left Right up , Donald Down I , Recall that in Cooper et ( 1989 ) Battle of the Sexes experiments with Homo sapiens , when one of the two players is allowed to communicate with the other player ( there is communication ) the players coordinate their choices 96 of the time ! However , with simultaneous communication between the two players , they coordinate only 42 of the time ! Explain what happened . We demonstrated how to solve for the Penalty Kick game equilibrium . Suppose you were new to the game of soccer ( or football ) and assigned to play the goalie position . After watching the following YouTube video , what strategy might make the most sense for you to adopt on penalty kicks BEHAVIORAL ECONOMICS PRACTICUM 343

watch ?

The map below identifies ( with red markers ) the locations of gas stations in Salt Lake City , Utah ( Utah capital city ) Do these gas station locations depict a pure strategy equilibrium for the Game ?

Explain . Em ' Clark i ' Chevron Salt Lake SUNBURST . Botanical ( Shell Smith Fuel Chevron Source Google Maps 12 . In this chapter , we learned that when an individual acquires private information about something , this added information does not 344 ARTHUR

13 . necessarily make the individual better off . In particular , when an individual ( say , Player ) acquires private information about something of common interest to both himself and another individual ( say , Player ) and Player knows Player has acquired this private information , Player could actually be made worse off as a result of Player changing her strategy in response to the fact that she knows Player now has additional information . Whew ! Can you think of a example where the acquisition of private information actually makes the individual worse off ?

For inspiration in formulating your answer , consider watching this trailer of the 2019 movie The Farewell watch ?

Analyze this excerpt from the British game show Golden Balls in the context of something you have learned about in this chapter watch ?

This question demonstrates how one player can turn a disadvantage encountered in a game into an advantage by making a prior public announcement and thereby transforming the game into a game . To show this , suppose the players payoff matrix is as shown below . a ) Determine this games analytical equilibrium in BEHAVIORAL ECONOMICS PRACTICUM 345

pure strategies ( its ) Explain why Player is not happy with this equilibrium . Suppose Player decides to preempt this games equilibrium outcome by announcing what his effort level will be before Player decides on her effort level . Describe this new game in its extensive form ( in the form of a decision tree ) Using dominance , identify the equilibrium for this game . Did Player preemptive announcement work to his advantage ( did Player give himself a advantage ) Explain . Explain why Player preemptive announcement does not necessarily have to be considered credible by Player to have its beneficial effect on Player . Low Effort High Effort Low Effort I , Player High Effort I I 15 . In 1944 , the Allies were planning their liberation of Europe while the Nazis planned their defense . The Allies were considering two possible landing beaches of Normandy or Pas de Calais . Calais was considered more difficult to invade but more valuable to win given its proximity to 346 ARTHUR

16 . Belgium and Germany . Suppose the payoff matrix facing the Allies and the Nazis is as depicted below . a ) What type of game does the payoff matrix represent ?

Calculate the games strategy equilibrium . Is this equilibrium consistent with what actually occurred in 1944 ?

Explain . Calculate the Allies expected payoff from following its equilibrium strategy . Nazi Defense Normandy Calais Normandy en , so 80 , so Allied Landing Calais 100 , Suppose two players are involved in a game where one of the players payoffs ( both when she cooperates and deviates ) are much larger than the other players ( because the player with the larger payoffs ( say , Player ) is a much larger producer than Player ) The payoff matrix for this game is provided below . Determine this games analytical equilibrium . Comment on this equilibrium in relation to the standard analytical equilibrium obtained in the Prisoner Dilemma . BEHAVIORAL ECONOMICS PRACTICUM 347

Cooperate Deviate 48 , 24 Deviate 60 , 12 40 , 16 Media Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure Chapter and 2000 American Economic Association is licensed under a All Rights Reserved license Figure Chapter and 2000 American Economic Association is licensed under a All 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 Figure ( Chapter ) Arthur is licensed under a BY Attribution license 348 ARTHUR

Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure ( Chapter ) Arthur is licensed under a BY Attribution license Figure 10 ( Chapter ) Arthur is licensed under a BY Attribution license Game Figure Arthur is licensed under a BY license Is More Information Always Better Figure Arthur is licensed under a BY ( Attribution license Is More Information Always Better Figure ( second figure ) Arthur is licensed under a BY ( Attribution license Weakest Link Figure Arthur is licensed under a BY license Weakest Link With Local Interaction Figure Arthur is licensed under a BY ( Attribution license Weakest Link With Local Interaction payoff matrix Arthur is licensed under a BY Attribution license Weakest Link With Local Interaction eight player schematic 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 BEHAVIORAL ECONOMICS PRACTICUM 349

is licensed under a BY Attribution license Figure for Study Question ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question 12 ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question 14 ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question 15 ( Chapter ) Arthur is licensed under a BY Attribution license Figure for Study Question 16 ( Chapter ) Arthur is licensed under a BY Attribution license 350 ARTHUR