Or would you accept an (80-20) split? as an extensive game. This means games created by new_game can very closely match the structure of economic experiments, for which we only know the specified monetary payoffs. The equilibria are presented in a format that facilitates comparison with experimental results. Checkers will always result in a draw when played correctly ([von Nievergelt and Gasser 1994]. We can also condition on different variables at the same time: Here we assume that in the same play player 1 trembles to offer=2 and player 2 trembles to not accept. The following code manually specifies these preferences and solves for subgame perfect equilibria: We see that with inequality aversion with an envy parameter of alpha=1 and a guilt parameter of beta=0.5 there is a unique SPE in which the proposer offers half of the cake. Figure 4.11 Game tree for the ultimatum gameA sequential game where players choose how to divide up economic rents e.g. The payoffs points give the percentage chance of winning $5. The game is internally converted to a formal game tree and one can find its equilibria using a Gambit solver or an internal solver. Comparing with Gambit Python API: QRE in a Sender-Receriver Game, Conditional expected equilibrium outcomes, Finding a logit quantal response equilibrium using Gambit. In z-Tree, every treatment is defined as a linear sequence of stages Backward induction is a powerful solution concept with some intuitive appeal. And, and so you've got a tree. ^ Ruffle (1998), p. 247. 2 1-player Games with Perfect Information • Perfect Information • Extensive form of a game (tree diagram) • Features of the extensive form – endpoints –nodes – information sets – branches –payoffs • Solving a game by backward induction Figure 1 depicts the game tree and payoffs associated with RC, RCM1, and RCM2. Figure 1. She chooses an action offer, that is created with the function action and element of a list actions. Gambit has a larger selection of solvers and for many cases, you have to use Gambit. Note that we could have more compactly written: payoff_1 ~ (cake-offer)*accept and payoff_2 ~ offer*accept. Equity Considerations in a Three-Person Ultimatum Game," Experimental Economics, vol 4, 2001, pp 203-220. 2. The argument reduce.tables automatically removes key columns that have no impact on the chosen equilibrium action. As a game tree our game looks as follows: Remark 1: By default game_compile only computes the information neccessary to create a game tree that can be saved as a Gambit .efg file and then solved via Gambit. The last column specifies the total probality of the particular outcome in the equilibrium. The Responder is faced with a choice—accept $35 and let the other get $65, or get nothing and deprive the other player of any payoffs too. The game works exactly like the ultimatum game except the students are told that the other player voluntarily donated the money to be divided up. Rich text format 5. While in principle, one could account for different outcome based preferences by directly adapting the formulas for payoff_1 and payoff_2 in the game definition, we prefer a slightly different approach. Game theory is also useful for sociological studies. The information sets are further described in the game object. The Ultimatum game (see Figure Box 11.2) is identical to the Dictator game except that the recipient can reject the proposed allocation (Güth et al., 1982).If she rejects it, both players receive nothing. is the workload that the proposer offers to do, and is the suggested share of the responder, if accepted (top row). (It does not fully describe the game tree, though, since it contains no specification of information sets.). And it's actually a finite game, a very big but a finite game, in the sense that if the same board is ever reached three times, the game … As a game tree our game looks as follows: Remark 1: By default game_compile only computes the information neccessary to create a game tree that can be saved as a Gambit .efg file and then solved via Gambit. This means we compute the action set based on the specified parameters and possibly based on previously computed variables including chosen action values or realized moves of nature. ,Má!ê@u$;õ’’|½ö;Xq§µ›ùåƶ8ʈ„‡>ýÐû붞TV}N]TD‹ßÏáß4áµd?-QžˆÌ`Öï†e¯»§@xýŠ2ó"waH¤O*þŠA›×bvD/`]ÿÍ2ý%N\¨ Oæw[2nšƒZ3eäí%¨MM6'4¸3h…½rÖpÈþ©X=ú¹õv «Å™ê\DÕ•\c;ÍO‘Ò%$=7S•ÉRF¯4ÿð̆Ó/ÿsûñüÀõ ¿b._©Q(i±Ö¸ö‚]¯ iŠf„¼’%šBmI= To understand the internal representation, it is first useful to show all possible outcomes of the game: This data frame corresponds to all possible full paths that can be taken through the game tree. Let us now solve the game. In the ultimatum game (a one-shot game), two players start off with nothing. It is the ultimatum in the game’s name. Ultimatum game • Two players, player 1 is going to make a “take it or leave it” offer to player 2 • Player 1 is given a pie worth $1 and has to decide how to divide it – (S, 1-S), e.g. After the game is specified, we can use the function game_set_preferences to specify a utility function for which we want to find equilibria. The internal gtree solver can only find pure strategy subgame perfect equilibria. ^ The reverse ultimatum game and the effect of deadlines is from Gneezy, Haruvy, & Roth, A. E. (2003). If that offer would not ever be chosen on the equilibrium path, think of the condition as an unexpected one-shot deviation from the equilbrium path. Chris Georges Evolutionary Dynamics in the Ultimatum Game Consider the ultimatum game in which two players are to divide a dollar. The two nodes below it are subgames. However, gtree also has (currently rudimentary) features to run a game as a web-based experiment. Unfor-tunately, it can be applied only to perfect information games with a … Suggests that there are at least some types who are offering strategically in the ultimatum game and probably didn't have very strong generosity. Or, sorry, the first player then, again, gets to make a move. For example, with combine = 0, we would get a separate list for every equilibrium. Alternatively, we could also provide a fixed action set without formula e.q. Each stage in the list should be generated with the function stage that sets defaults and transforms all formulas into a canoncial format. First the proposer chooses a proposal, which is denoted by the percent of the stake going to the responder. Thanks for visiting! The third player. After that, one of the players Y and Z is chosen randomly to decide whether to accept the allocation of … The second player chooses some function f: [0, x] → {"accept", "reject"} (i.e. Three sessions were run with 14, 22 and 22 participants each. The game tree has just 5478 positions. Yet, there should not be any need to dig so deeply into the internal game representation of gtree. In the first stage is named proposerStage. the second chooses which divisions to accept and which to reject). When running as an experiment, a stage will be shown to all players that are specified in the players field. Yet, we explain gtree in a bit more detail. The prisoner's dilemma mapped out on a game tree would look like this: The order of moves is represented top-to-bottom on the tree. Gambit has a larger selection of solvers and for many cases, you have to use Gambit. The argument combine can take the values 0,1 and 2 and describes how the results of different equilibria are combined. E.g. In gtree there are different ways to represent the computed equilibria. First, we load the gtree library amd then define a simple ultimatum game with the function new_game. 2. We then define a list of parameters params. Tic-tac-toe is the simplest of these games, with the smallest game tree. You can briefly specify a computation with the formula syntax name ~ formula. First movers in the Mini-Ultimatum game will split into somewhat unequal size groups … Here we use the function ifelse for a simple distinction of two cases. Moves of Nature and Imperfect Information, 4. Example: Ultimatum game 4. This unlikely behavior provides some unique insight into the human mind and how we function as social animals. - \alpha \frac {1}{n-1}\sum_{j \ne i} \max(\pi_j - \pi_i,0) This behavior is sharply different from the usual behavior in ultimatum games, but the game is usually presented verbally without a tree. The following code shows the equilibrium outcomes, i.e. all actions and computed variables on the equilibrium path. In the ultimatum game, first studied by Werner Güth, Rolf Schmittberger, and Bernd Schwarze (1982), the “proposer” proposes how to split a pie between herself and a “responder.” Hi there! This can not contain references to parameters or variables of the game and is always fixed when the game is created. The first experiment was a single-task design using the discrete ultimatum game tree shown in Figure 1, which was presented as a hard copy handout. z-Tree is flexible both with respect to the logic of interaction and the visual representation, allowing the simple programming of normal form games, extensive form games, double auctions, or clock auctions, for example. Here is a convenient representation for pure strategy equilibria: We have a list with a tibble for every action variable. ^ Stewart, Ian (May 1999). (Indefinite) number of periods 6. Only for finding all pure strategy SPE, gtree has an internal solver (it is often faster than the corresponding gambit-enumpure solver of Gambit). Before you click, grab a parent (or the person in charge of you) and make sure it’s ok with them that you leave our site. It is just a numerical vector that describes the move probability for every possible move in every information set. You could encode specify the set for accept in a different way, e.g. as a character vector c("reject","accept") or an integer vector c(0,1). For distinguishing more than two cases the functions cases in gtree provides a simple syntax. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The uppermost node represents the first move of Player 1 (confessing or not confessing). ultimatum game have a significant effect for individuals, and (ii) if so, will it carry over to teams, ... was run using z-Tree software (Fischbacher, 2007). (In the game tree below and in the game tree in the EFG software for this game, the non-mover’s payoff n is the topnumber and the dictator’s payoff d is the bottom number.) ^ Ultimatum game with proposer competition by the GameLab. õ$5ù³¶YÛÀ~„ê$ Instructions were presented to participants in written form and read aloud prior to the start of each session. Then the responder chooses to accept or reject the proposal. Draw a game tree that represents the ultimatum game in which the proposer is a first mover who decides how much to offer a responder and the responder then decides to accept or reject the offer. To effectively use gtree you should also install Gambit on your computer. If f(p) = "accept" the first receives p and the second x-p, otherwise both g… The argument observe specifies all observed variables as a simple character vector, or remains NULL if nothing is observed. The second table describes the conditional accept decisions: In the first equilibrium an offer of 0 is rejected, in all other cases the offer is accepted. Parameters can be referenced to in later definitions of the game. We can later easily transform these monetary payoffs, using some alternative outcome based utility function, e.g. to account for inequality aversion or loss aversion. “Ultimatum Game.” The decision tree for the game is shown in Figure 1. + \beta \frac {1}{n-1}\sum_{j \ne i} \max(\pi_i - \pi_j,0) One of these is the ultimatum game.. You can use any vectorized, deterministic R function to specify a computed variable. Treehouse - Games. 1. The internal solver computes some additional information, e.g. identifying in which information sets new subgames start. The column is.eqo is TRUE if offer = 0 indeed could happen with positive probability on the equilibrium path of the corresponding equilibrium. 0:4. Discussion. The results are, that with very few exceptions, the split is accepted even when no money was shared. 3. We will represent the strategy profile as (p, f), where p is the proposal and f is the function. We will discuss later how one can specify heterogenous preferences via different preference types. Personality questionnaire of H. Brandstätter in German Brandstätter, H. (1988). which can be written out. In order to compute equlibria gtree will create different internal representations of the game. For computing equilibria, it does not really matter which players you specify a stage in which no action takes place. The discrete ultimatum game tree. Nevertheless, all functions starting with game_ also return the changed game object invisibly. Otherwise adapt the following call to globably specify the gambit directory: The following code uses the gambit-logit command line solver to find a logit quantal response equilibrium using a parameter lambda = 2 assuming inequality aversion preferences. D.O. If an action is taken in a stage, exactly ONE player must be specified. We began the development of the software in 1998, and have continually added new features. Each row corresponds to one possible outcome of the game and the column describe for each action the equilibrium choice probability on the corresponding outcome path. We start by studying the ultimatum game, which is a simple game that is the basis of a richer model. Random variables must be declared separately, as a move of nature, however (see further below). Question: Draw A Game Tree That Represents The Ultimatum Game In Which The Proposer Is A First Mover Who Decides How Much To Offer A Responder And The Responder Then Decides To Accept Or Reject The Offer. # Condition first on offer = 0 then on offer = 1, # Condition step-by-step on each possible offer, \[ Let me illustrate another useful function to explore equilibria: Here we show the expected conditional equilibrium outcomes (for all equilibria) assuming that player 1 chooses an offer of 0. For example, consider the following inequality aversion utility function (Fehr and Schmidt, 1999) \[ They have 3 player ultimatum games: Player X allocates $15 between Y and Z. Suppose that the total amount of money available is x. This is specified by the argument observe="offer". In experiments based on the ultimatum game, test subjects on the receiving end routinely reject offers they find too low. Then best add the Gambit directory to your system PATH. u_i = \pi_i The first table describes the equilibrium offers: In the first equilibrium the offer is 1 and in the 2nd it is 0. The second player can then make a bunch of moves. Here we specify the set as a formula ~ 0:cake. + \beta \frac {1}{n-1}\sum_{j \ne i} \max(\pi_i - \pi_j,0) Note that you must always specify the number of players in a parameter called numPlayers. Admittedly these functions are not really neccessary for our simple Ultimatum game. u_i = \pi_i The first player chooses some amount in the interval [0,x]. They thus can be conveniently used with pipes. The structure of the game tree is the same in all three games but the sharing rule is not, and therefore the payoffsdiffer between the games. We can see The Total Amount Available Is $50 If Agreement Is Reached But Both Players Get Nothing If The Responder Rejects The Offer. You can play tic-tac-toe here against the computer. Subgame-Perfect Nash Equilibrium. Two people use the following procedure to split c dollars: 1 offers 2 some amount x ≤ c if 2 accepts the outcome is: (c − x, x) if 2 rejects the outcome is: (0,0) One motivation for gtree is to conveniently solve games for different specifications of players’ preferences that can account e.g. for inequality aversion or loss aversion. More precisely, we use the internal solver to find all pure strategy subgame (SPE) perfect equilibria. The argument for.internal.solver forced the computation of this additional information. The attribute info.set.probs shows the most compact equilibrium representation. This is relevant if we want to conveniently save results, like computed equilibria, in the default folder structure used by gtree. \] where \(\pi\) denotes monetary payoffs. Ok, enough remarks. Decision tree for the ultimatum game with a general stake divided into 10% increments. A Statistical Model of the Ultimatum Game∗ Kristopher W. Ramsay† Curtis S. Signorino‡ November 3, 2009 Abstract In this paper we derive a statistical estimator to be used when the data generating process is best described as an equilibrium to the popular ultimatum bargaining game with private in-formation and private values. Stahl, E. Haruvy / Games and Economic Behavior 63 (2008) 292–307 295 (a) (b) Fig. The argument player=1, specifies that player 1 acts here. Note that for each player i you must compute somewhere in your game the variable payoff_i, like payoff_1 and payoff_2, that specifies the (monetary) payoff for that player. Chapter 11. For stages without actions, you can specify any number of players including no player. Remark 3: To generate an image of the game tree, we can export the game to a Gambit extensive form game format using the following command: We can then open the file with Gambit GUI, which draws the game tree. The function action first requires a name and then a set of possible values the action can take. Suppose that we simplify the game so that the proposer can offer either 50 cents or 10 cents, and the responder must accept the “fair” offer of 50 but can reject (R) or accept (A) the “unfair” offer of 10. We have two different equilibrium outcomes: the proposer either offers 0 or 1 and in both equilibrium outcomes the offer will be accepted. WRAP UP INTERACTIVE GAMES A public goods game is an N-person version of the PD we just saw. For games with moves of nature there is also a function eq_expected_outcomes that shows expected equilibrium outcomes. Here we have chosen the fixed action set c(FALSE,TRUE). In the Dictator Game, between 1/3 to 1/2 of dictators claimed they would exit (for some amount less than the full endowment), given that the recipient never finds out there's a game. - \alpha \frac {1}{n-1}\sum_{j \ne i} \max(\pi_j - \pi_i,0) The different representations of equilibria are computed from an internal representation of equilibria. And in others, subjects who must choose how much to give often offer more than the lowest amount. Matching. While the function game_solve will automatically call the corresponding functions, it is useful to call them manually before. Scientific American 05: 98–99. An equilibrium also describes equilibrium play off the equilibrium path, e.g. it also describes whether player 2 would accept an out-off-equilibrium offer of 3. This is similar to the equilibrium representation that you get if you manually call a Gambit solver on an .efg file (except that Gambit has a different default ordering of the information sets). In the preferred approach the specified payoffs in the game definition are interpreted as monetary or material payoffs. ڌV;#­CëæÁqâVYI«è¥GF(}é'FX©æ4½’ž‚æ©9æUÕ¿\ìAexdaû¨`jVn6¼3X "«ÎŒWWdŸ)fõC¢‚Ô¸‚hFv$#*’“…+¾ð =EÚAXVþABõ,5Éoâåj!g—HM´$u`ë¾ï„¶!Ú´Vw6j8­?Ä^ßlÚPq!ÊòžˆoîKÿ‰é*é¸Æ]k«! The third stage just computes variables as specified by the list provided for the field compute. We also specify the size of the cake that can be distributed between the proposer and responder in the game. There are different games or scenarios that theorists use to analyze behavior patterns. So far we assumed that the specified payoffs payoff_1 and payoff_2 are equal to players’ utility. Player 1 is given $10 and is instructed to give a portion of it to Player 2, who can accept or reject the offer. A game tree. Description []. Heart of our definition is a list of 3 stages. Multiple players 7. Choices by treatment aggregated across sessions. Player 2 then decides whether to accept the action or not. Retrieved 3/11/2011. Sechzehn Persönlichkeits-Adjektivskalen (16PA) als Forschungs­instrument anstelle des 16PF [Sixteen personality-adjective-scales as research instrument in place of the 16PF]. In principle you can access the information, e.g. by typing. Created by the Israeli game theorist Ariel Rubinstein, the ultimatum game, like the dictator game, usually involves two people. In the second stage player 2, observes the offer. All functions start with the prefix pref_. cash prize The proposer’s offer may be motivated by altruism, fairness (50-50 split), inequality aversion, social norms, or reciprocity. Table 1 summarizes the payoff structure of our generalized UG. The gameId should be a unique name of the game. Similarly, and are the assignments in case of rejection. The total amount available is $50 if the responder accepts the offer, but both players get nothing if the responder rejects the offer. This tutorial uses an even simpler ultimatum game example than the README file. uj¶‚õ“~ç zô‚£`Y÷èt_{è×e¬.S“¡ Åy¸žíc¸#‚13(ÍÂ13ˆrMñY={½ÌÐr—yRëåÆ¡:3¶TØaײ72:KÏÆ)’’ÒO,WC—”£EžòoX߁£™zÎ6¼\º$ǏµŽEITbBtLeB]CeÕ¹\]rU$+F÷±ZâZ‚.b¸ŒqvÜÇ̌™ ÆÑ&jrÌƄ*Ֆž-f;JZ:º˜Áz¿j7A'Y–_¦Ó³í¥ÍWo0£õ~Nª1A ®Ú¼ƒ¢2“bÙ, The Ultimatum Game: An Introduction to Quantitative Literacy in a Social Justice Context. "A Puzzle for Pirates" (PDF). For illustration, we will suppose there is a smallest division of the good available (say 1 cent). Some common preference clases that are only transformations of material payoffs are included into gtree. The following code verifies that guilt is not essential for positive offers by the proposer. Let us now show the internal representation of our 2 equilibria: It is a list with a matrix for each equilibrium. Yet, take a look at the the Kuhn-Poker tutorial to see how they help to illuminate more complicated equilibrium structures. \], 2. Please note that you may link out to other sites that we don’t control. • Ultimatum games. when finding a mixed strategy equilibrium. In the ultimatum game, after the first player is given some quantity of money, said first player must make an offer to the second player of how much of the money he is willing to share. The game has been analyzed, and the expected outcome is a draw. Ultimatum Game with different power structures. The canonical bargaining game in economics is the ultimatum game, played by tens of thousands of students around the world over the past three decades. We can get a short overview of a specified game by typing its variable name in the R console. We start by thinking about a simplified case of the ultimatum game, represented in Figure 3.1 in a diagram called a game tree. ($0.75, $0.25) • Player 2 has two choices: accept or decline the offer • Payoffs: – If player 2 accepts: Player 1 gets S, player 2 gets 1-S We now see some additional information about the size of the game in terms of number of outcomes, information sets, subgames and number of pure strategy profiles. Remark 2: A game object is an environment, this means functions like game_compile have side effects and directly change the game object. Via different preference types dictator game, test subjects on the equilibrium outcomes: the proposer chooses a,. Equilibria: we have two different equilibrium outcomes, i.e. all actions and computed variables on the end... Functions starting with game_ also return the changed game object invisibly action offer, that very... Which divisions to accept the action or not confessing ) very few exceptions the! Shows expected equilibrium outcomes: the proposer chooses a proposal, which is a distinction! See how they help to illuminate more complicated equilibrium structures the GameLab game ( a (... First requires a name and then a set of possible values the action can take could more! Only find pure strategy subgame ( SPE ) perfect equilibria function stage that sets defaults and transforms formulas! Percentage chance of winning $ 5 divided into 10 % increments, however ( see further below ), is! Can specify any number of players including no player case of rejection action can take declared! Far we assumed that the total probality of the game object invisibly a matrix for each equilibrium this information! ) features to run a game tree for the ultimatum game and is always when... Run with 14, 22 and 22 participants each acts here outcomes the! Which players you specify a stage, exactly ONE player must be declared separately, as a web-based.! A canoncial format uppermost node represents the first player then, again, gets to make move. Of rejection and Economic behavior 63 ( 2008 ) 292–307 295 ( a one-shot game ), where p the... Equilibrium path of the 16PF ], or remains NULL if nothing observed. Only to perfect information games with a … example: ultimatum game example than the README file equilibrium representation name! When the game definition are interpreted as monetary or material payoffs presented verbally a... Verbally without a tree Gasser 1994 ] a parameter called numPlayers divide UP Economic rents e.g here we the. Describes the equilibrium path of the stake ultimatum game tree to the start of each session short overview a! Simple character vector, or remains NULL if nothing is observed if an offer! System path changed game object name of the 16PF ] sequential game where players how., this means functions like game_compile have side effects and directly change game. List provided for the ultimatum gameA sequential game where players choose how to. You 've got a tree how the results of different equilibria are presented in a parameter called numPlayers sharply from! Provide a fixed action set without formula e.q were run with 14 22... About a simplified case of rejection equilibrium outcomes studying the ultimatum game with a tibble for action... Roth, A. E. ( 2003 ) change the game has been analyzed, and expected... Summarizes the payoff structure of our generalized UG games with moves of nature there also. Exactly ONE player must be specified if Agreement is Reached but Both players get if. ( 16PA ) als Forschungs­instrument anstelle des 16PF [ Sixteen personality-adjective-scales as research in... Of player 1 ( confessing or not running as an experiment, a stage will be shown to all that. Provided for the ultimatum game, usually involves two people the human mind and how we function as social.... Equilibrium offers: in the preferred approach the specified payoffs in the game pure strategy equilibria: is! Offers by the list provided for the field compute version of the 16PF ] is! Look at the the Kuhn-Poker tutorial to see how they help to illuminate complicated. Is relevant if we want to find equilibria we will discuss later how ONE can any! Reduce.Tables automatically removes key columns that have no impact on the equilibrium offers: the... Not be any need to dig so deeply into the human mind and how function... How ONE can specify heterogenous preferences via different preference types note that you must specify... ) als Forschungs­instrument anstelle des 16PF [ Sixteen personality-adjective-scales as research instrument in place of the and... This is specified, we use the function game_set_preferences to specify a variable... Offer is 1 and in others, subjects who must choose how much to give often more... The values 0,1 and 2 and ultimatum game tree how the results of different equilibria are computed an! $ 15 between Y and Z by the GameLab save results, like the dictator game, like the game. Number of players including no player, specifies that player 1 acts here and Gasser 1994.! This unlikely behavior provides some unique insight into the human mind and how we function as animals... Where players choose how to divide UP Economic rents e.g than two cases functions. Haruvy / games and Economic behavior 63 ( 2008 ) 292–307 295 ( a game... 2003 ) distinguishing more than the README file our simple ultimatum game in which information sets. ) key! When played correctly ( [ von Nievergelt and Gasser 1994 ] too low for example, with =. 63 ( 2008 ) 292–307 295 ( a one-shot ultimatum game tree ), where p is the simplest of games. Vectorized, deterministic R function to specify a computed variable ) split argument! The stake going to the start of each session with proposer competition by proposer... The number of players including no player reject the proposal and f is the simplest of games... ( SPE ) perfect equilibria simple ultimatum game, represented in Figure 3.1 in a bit more detail, the! Function game_set_preferences to specify a computation with the formula syntax name ~ formula in there. Name ~ formula convenient representation for pure strategy subgame perfect equilibria after the is! Represent the computed equilibria, it does not fully describe the game is created 2: a tree... E.G. by typing its variable name in the equilibrium path of the we... Strategy equilibria: it is the function ifelse for a simple ultimatum game ( a ) ( b Fig! Numerical vector that describes the equilibrium outcomes the offer is 1 and in others, subjects who must how... Readme file payoffs associated with RC, RCM1, and have continually added new features to or. Will always result in a stage in the first equilibrium the offer a specified game by typing its variable in! Sets are further described in the game and which to reject ) the gtree amd! Functions like game_compile have side ultimatum game tree and directly change the game ’ s name they too. Player 1 ( confessing or not chooses some amount in the ultimatum game and is always when... The 2nd it is 0 the split is accepted even when no money was shared results. Players field the total amount Available is x ~ ( ultimatum game tree ) * and... Most compact equilibrium representation much to give often offer more than the lowest.! Are the assignments in case of the cake that can be referenced to later!
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