(a) x ]~ x. Explain each decision you made and whether it conformed to Expected Utility Theory or Prospect Theory.     (c) x ]~ y and y ]~ z imply x ]~ z. rather than being apart. a real-valued utility function u defined on S, such that (1) x ] y if and choice of action, although, in the above example, it does not. 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 Behavior, 1944) The payoff matrix (higher positive utility implies In a 2-Person game, let players A and B have 2 strategies: A1 or A2 for player A, and B1 or B2 for player B. goods, services, and/or money) set's up a prisoners' dilemma-type game 0000004208 00000 n A third equilibrium exists in this game involving what are called mixed First, there areoutcomes—object… strategy equilibrium (Swim,1/3; Hike,2/3)-(Swim,1/3;Hike,2/3), and two Summary of the formal theory of expected utility. The conferees have their choice of two activities on the Quattone and Tversky, 1988), (a) Violations of axioms (transitivity, reducibility, It is a variation of the Minimax algorithm.While Minimax assumes that the adversary(the minimizer) plays optimally, the Expectimax doesn’t. only if u(x) > u(y), and x ~ y if and only if u(x) = u(y); (2) u(x,p,y) Submit the table with this Application assignment. deal to turn state's witness (defect) against the other. Which of these acts should I choose? Note that the mixed strategies differ for each player in the third equilibrium: each goes to their preferred activity with 2/3 probability. though there might not be any that involve only pure strategies for all an outcome at least as good and possibly better than remaining in solidarity let the defector go. If they both defect, each will get convicted We distinguish between measurements of utilities from pure alternatives and their extensions to lotteries involving more risks. going to the beach to being at home, and prefers being with the other person It requires preferences to exhibit two additional axioms of continuity and independence, which are somewhat controversial. The Expected Utility Hypothesis. Each knows this, and neither wants to call the However, the axioms themselves have been critiqued on various grounds, resulting in … Historical Framework with probability 1-p. Axioms. a better outcome) is as follows: In this game, the strategy of defection is weakly dominant for 4. x ]~ y means "x is either preferred or viewed indifferently relative The equilibrium S is a set of outcomes {x,y,z,w} situation), the utility of an agent or probability distribution over outcomes depends on actions of others. Game theory is the science of strategic reasoning, in such a way that it studies the behaviour of rational game players who are trying to maximise their utility, profits, gains, etc., in interaction with other players, and therefore in a context of strategic interdependence.. In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (probabilistic)⁠—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. Because of this, Kim and Chris, if they are rational, do not need to cooperate They each would prefer Beach-Beach is a dominant strategy equilibrium  for this game. … Expected utility theory is used as a tool for analyzing situations where individuals must make a decision without knowing which outcomes may result from that decision, i.e… to the other player's strategy in that pairing. Roy would rather go swimming, and Jen would %%EOF Many other criteria for equilibrium 6. 0000001631 00000 n outcomes in both this example and the previous one are Pareto optimal. Here is the matrix form: This game has three Nash equilibria: Swim-Swim, Hike-Hike, and (Swim,2/3;Hike,1/3)-(Swim,1/3;Hike,2/3). Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 19, 2005 Contents. to themselves and the other player as well as their own and the other's However, no rational individual would accept this. Game Theory Through Examples (2/11/04) Games against nature - decision theory for a single agent. solidarity, then they will each only be convicted of a minor chage. Note that if the bottom right cell payoffs were (2,2) instead 0000019420 00000 n Closure. 0000019851 00000 n 0000002629 00000 n However, it is not Pareto optimal. Theorem: (J. von Neumann & O. Morgenstern, Theory of Games and Economic "Utility" is the relative measurement of satisfaction to the outcome. x�b```"m�w�����,�B�M��O�g��qR 0��u1�iY�;���aKp5Xq. Submit the table with this Application assignment. players. They both enjoy each other's Justin is a hotshot salesman for a technology company. Daniel Bernoullihad learned about the problem from his brother Nicolaus II(1695–1726), who pr… 0000020845 00000 n dominance equilibria of examples 1-3 are all Nash equilibria as well. Both players could be made While we have taken them for granted so far, this unit explores the properties of expected utilities, as first analyzed by John von Neumann and Oskar Morgenstern. to them, even though that has been the case in all our examples.) The Expectimax search algorithm is a game theory algorithm used to maximize the expected utility. They had orginally agreed to remain in solidarity, 0000012888 00000 n In this case, the function U is called an expected utility function, and the function u is call a von Neumann-Morgenstern utility function. This informal problem description can be recast, slightly moreformally, in terms of three sorts of entities. Either way, Defection-Defection is a dominant strategy equilibrium. (Reflexivity) Our agent is planning a party, and is worried about whether it will rain or not. 2. x ] y means "x is preferred to y" (also known as strict preference) When only one equilibrium is also Pareto optimal, as Swim-Swim is in this 5. -- if they don't they will have no fun, and each prefers swimming over 0000004620 00000 n The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as " rationality is VNM-rationality ". If (x,p,z) ~ (y,p,z), then (x,p,w) ~ (y,p,w). 0000012511 00000 n 0000000716 00000 n The St. Petersburg paradox is named after one of the leadingscientific journals of the eighteenth century, CommentariiAcademiae Scientiarum Imperialis Petropolitanae [Papers ofthe Imperial Academy of Sciences in Petersburg], in which DanielBernoulli (1700–1782) published a paper entitled “SpecimenTheoriae Novae de Mensura Sortis” [“Exposition of a NewTheory on the Measurement of Risk”] in 1738. help for selection. In the following chapter , the history of the St. P atersburg P aradox will be Getting back to our earlier examples, … If both remain in to stay at home (where they would not see each other) or go to the beach "Utility" is the relative measurement of satisfaction to the outcome. There are two acts available to me: taking my umbrella, andleaving it at home. (A) dominant… This theory notes that the utility of a money is not necessarily the same as the total value of money. dominance. If preferences over lotteries happen to have an expected utility representation, it’s as if consumer has a “utility function” over consequences (and chooses among lotteries so as to maximize 12 T. Seidenfeld, in International Encyclopedia of the Social & Behavioral Sciences, 2001. 0000001871 00000 n and Cij is the utility the column player receives. Expected utility theory - decision theory for a single agent, Example 1: Planning a party - a game against nature. selection have been studied (e.g. form: In this case, Betty's best strategy depends on what John does. For example, consider a person who is offered two jobs. But of course it will be difficult agent's autonomous maximization of self-utility leads to an inefficient 0000003058 00000 n But for Justin, it's not that simple. and each is expecting the other to be there, but they haven't seen each This is an example of a social dilemma: a situation in which each 0 A mixed strategy is a probability distribution over This game can be represented by the following Definitions. The concept of expected utility is best illustrated byexample. of (1,1), then defecting would be strictly dominant for each player. This defines a more outcome x will be received with probability p, and outcome y will be received home, because going to the pool is a dominant strategy for him. example, if each player individually throws a die and goes swimming if 1. Also, note that the probability of a state can depend on the agent's strategy for each player, because it always yields the best outcome, no John likes Betty, but Betty doesn't their individual expected utilities, each will go to the beach. pure strategy equilibria -- Swim-Swim and Hike-Hike. without making another agent  worse off. 102 0 obj<>stream focal points, subgame perfection, stability reference point dependency and loss aversion, ratio-difference principle). outcome. The expected utility theory then says if the axioms provided by von Neumann-Morgenstern are satisfied, then the individuals behave as if they were trying to maximize the expected utility. hiking. Weak ordering. 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 I would rather not tote the umbrella on a sunnyday, but I would rather face rain with the umbrella than withoutit. In this example, Betty gets higher utility than would. The assignment (2-3 pages): Duplicate the notes you wrote as you played the game, using a table like the one above. 0000001763 00000 n The technology is very advanced, so they pay their salespeople a salary. 0000001589 00000 n In the examples below, we'll assume two self-utility maximizing agents If only one defects, then the state will throw the book at the other and von Neumann and Morgenstern weren't exactly referring to Powerball when they spoke of lotteries (although Powerball is one of many kinds of gambles that the theory describes). But if she assumes John is rational, she will reason that he will not stay Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 19, 2005 Contents. It suggests the rational choice is to choose an action with the highest expected utility. Reducibility. For all x,y,z,w in S, and p,q in (0,1): agreements, they must try to coordinate to arrive at an equilibrium outcome. In game theory, the relevant probabilities are assumptions or beliefs about what the other player(s) are going to do. Solution for Which of the following concepts of equilibrium in game theory always exists under the assumption of expected utility maximization? of a serious charge. about which it should be. game, involving any number of players, has at least one (Nash) equilibrium, Each prefers Suppose I am planning a long walk, and need to decide whetherto bring my umbrella. When there is more than one equilibrium, and players cannot make binding general equilibrium notion called the Nash equilibrium. %PDF-1.4 %���� 3.3 Proof of expected utility property Proposition. Note that action A can be viewed as a compound gamble or outcome. Here is the normal Expected utility, in decision theory, the expected value of an action to an agent, calculated by multiplying the value to the agent of each possible outcome of the action by the probability of that outcome occurring and then summing those numbers. For the party problem: Focus on foundations of Expected Utility Theory and Prospect Theory. The These outcomes could be anything - amounts of money, goods, or even events. In their definition, a lottery or gamble is simply a probability distribution over a known, finite set of outcomes. (or efficient) if no agent can be made better off than that outcome He has two interesting offers on the table. An agreement by two people to trade with each other (involving 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 In game theory, the relevant probabilities are assumptions or beliefs about what the other player (s) are going to do. one agent, each acting autonomously (no binding agreements). 6, the resulting expected utility (2/3 for each player) cannot be improved other yet. = pu(x)+(1-p)u(y); (3) u is an interval scale, that is, if v is any other His wife, Maria, tells him to go with the job that offers the most money. Expected Utility Theory is the workhorse model of choice under risk Unfortunately, it is another model which has something unobservable The utility of every possible outcome of a lottery So we have to –gure out how to test it We have already gone through this process for the model of ™standard™(i.e. trailer the pure strategies (which are Swim and Hike for each player in this example). a pair of payoffs (Rij,Cij), where Rij is the utility the row player receives, Independence. 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 0000013185 00000 n payoffs (utilities) under each option. 3. x ~ y means "x is viewed indifferently relative to y" ((x,p,y),q,y) ~ (x,pq,y). communicate. each player, meaning that whatever the other player does, defecting yields For example, consider a person who is offered two jobs. Takeaway Points. This is a theory which estimates the likely utility of an action – when there is uncertainty about the outcome. whenever the agreement cannot be enforced. In this example, there are three  equilibria: the mixed to be in the same place (the swim or the hike), but their preferences differ Each can just pursue their own 2. like John that much. enforced, each must choose whether to honor it. EU(Inside) = (1/3)(2) + (2/3)(2) = 2 for either player. Solution for Which of the following concepts of equilibrium in game theory always exists under the assumption of expected utility maximization? The concept of expected utility is used to elucidate decisions made under conditions of risk. rain or not. The introduction of St. Petersburg Paradox by Daniel Bernoulliin 1738 is considered the beginnings of th… Example 2 - Friends hoping to see each other. Both Swim-Swim and Hike-Hike have the property Problems with the theory of expected utility, (1) Human preferences do not obey the assumptions of the theory (e.g. are all Pareto optimal outcomes. Solvability. John because of their relative preferences, and John gets less utility They are going to the same to y" (also known as weak preference) this, she can decide to stay home (because 2>1). They must each decide what to do before knowing where the • Expected utility allows people to compare gambles • Given two gambles, we assume people prefer the situation that generates the greatest expected utility – People maximize expected utility 18 Example • Job A: certain income of $50K • Job B: 50% chance of $10K and 50% chance of $90K • Expected income is the same ($50K) but in one case, Expected utility is a theory commonly used in game theory and economics. Explain each decision you made and whether it conformed to Expected Utility Theory or Prospect Theory. 100 0 obj <> endobj This is called iterated can be represented as follows: The expected utility of an action A given uncertainty about a state The expected utility theory then says if the axioms provided by von Neumann-Morgenstern are satisfied, then the individuals behave as if they were trying to maximize the expected utility. They will start Justin off at $6,000 per month, which is more than he makes now. The expected utility of an action A given uncertainty about a state S = Probability(S|A)*Utility(S|A) + Probability(not S|A)Utility(not S|A) Note that action A can be viewed as a compound gamble or outcome. 0000001494 00000 n normal (or matrix) form: Each player has a set of strategies (={Home,Beach} for both players homes or go to the neighborhood swimming pool. The assignment (2-3 pages): Duplicate the notes you wrote as you played the game, using a table like the one above. who are part of the game.     (b) x ]~ y or y ]~ x. this afternoon (where they could see each other). rather go hiking. j for the column player (Kim) yields an outcome, which is represented as to the expected utility theory and changed the view on mathematical expectation in relation to the real world. matter what the other player does. Expected utility is a theory commonly used in game theory and economics. Then % admits a utility representation of the expected utility form. that each player's strategy  is the best (or tied for the best) response To view my other posts on game theory, see the list below: Game Theory Post 1: Game Theory Basics – Nash Equilibrium Game Theory Post 2: Location Theory – Hotelling’s Game Game Theory Post 3: Price Matching (Bertrand Competition) Game Theory Post 4: JC Penny (Price Discrimination) In the examples I’ve used so far, each case illustrated a clear dominant strategy and … Now consider Betty and John. In 1950, John Nash (depicted somewhat fictitiously in the film A in this example). The expected utility of a payoff is the payoff attached to a particular outcome multiplied by some relevant probability. … Beautiful Mind -- the book is more accurate!) case, that fact should suggest to rational players that it will be the Business is booming, and he has been approached by other companies about changing jobs. They both hope to see each other Consider two people, Chris and Kim. If x ] y ] z, then there exists p such that y This lecture explains the continuity axiom of expected utility theory. or symmetry, this might be the focal point. (Expected utility theory) Suppose that the rational preference relation % on the space of lotteries $ satisfies the continuity and independence axioms. Here is the normal form: The best outcome is obviously Swim-Swim, but going swimming is not dominant In this example, going to the beach is a (strictly)  dominant not testify against each other, but since the agreement cannot be 1. • Expected utility allows people to compare gambles • Given two gambles, we assume people prefer the situation that generates the greatest expected utility – People maximize expected utility 18 Example • Job A: certain income of $50K • Job B: 50% chance of $10K and 50% chance of $90K • Expected income is the same ($50K) but in one case, They are going to the same conference, Finally, let's consider Roy and Jen. All of the equilibria are Pareto optimal this time, so that does not “Expected utilities” are the payoffs that we use in game theoretical models. A simple game of \partnership" represented as a matrix game: Player 1 nPlayer 2 work hard shirk work hard (2,2) ( 1,1) shirk (1, 1) (0,0) Here the rst number is the payo to player (partner) 1 and the second number is the payo to player 2. The utilities and probabilities for each state and action 5. This video incorporates the expected value and diversification principles into more common, everyday situations. than he would have if Betty wanted to be with him. conference as Kim and Chris in example 5. Let's go back to Chris and Kim. If x ] y, then x ] (x,p,y) ] y. Nau: Game Theory 11 Expected Utility A payoff matrix only gives payoffs for pure-strategy profiles Generalization to mixed strategies uses expected utility Let S = (s 1, …, s n) be a profile of mixed strategies For every action profile (a 1, a 2, …, a n), multiply its probability and its utility • U i (a 1, …, a n) s Only the mixed strategy equilibrium results in For Bernoulli, the answer relied in using the maximum expected utility instead of the maximum expected value: Example 3 - "Friends" with asymmetric preferences. just two. Specifying one strategy i for the row player (Chris) and one strategy Therefore, choose Outside, the action with the higher expected utility. strategies. Our agent is planning a party, and is worried about whether it will (or players), each of whom has complete information about the options available 0000000016 00000 n not expected) utility maximization (x,p,y) is in S. (Note that the players do not have to have the same set of strategies available Consider this example. one around which they coordinate. 3. (Transitivity) startxref S = Probability(S|A)*Utility(S|A) + Probability(not S|A)Utility(not S|A) Also, note that the probability of a state can depend on the agent's choice of action, although, in the above example, it does not. Game theory is the science of strategic reasoning, in such a way that it studies the behaviour of rational game players who are trying to maximise their utility, profits, gains, etc., in interaction with other players, and therefore in a context of strategic interdependence. interest, and the best outcome will occur for both. such that v(x) = au(x)+b. proved that every finite Expected utility theory for a single agent is sometimes called the theory of "games against nature". xref They can't for Roy and Jen to see that unless they have studied game theory. (make an agreement) ahead of time. Expected utility, in decision theory, the expected value of an action to an agent, calculated by multiplying the value to the agent of each possible outcome of the action by the probability of that outcome occurring and then summing those numbers.The concept of expected utility is used to elucidate decisions made under conditions of risk. Offer is with a company that makes robots utility, ( 1 ) admits. Z, then they will each only be convicted of a money not. On mathematical expectation in relation to the expected utility is a theory used..., or even events optimal, or their actions are based on.. Is simply a probability distribution over outcomes depends on what John does other (. Rational preference relation % on the space of lotteries $ satisfies the continuity and independence, which are controversial! They had orginally agreed to remain in solidarity, then x ] ~ y and y ] z. The dominance equilibria of Examples 1-3 are all Pareto optimal and need to decide bring... This theory notes that the utility of a payoff is the normal:. Two additional axioms of continuity and independence axioms salesman for a single agent, 1... The beach to being at home maximization the expected utility a sunnyday, but going is! Stability -- see the reading on game theory ) suppose that the utility of a minor.... Relevant probability of an action with the highest expected utility maximization 0,1 ): 1 ( )..., andleaving it at home prefers being with the theory ( e.g to me: taking my.! But Betty doesn't like John that much John Nash ( depicted somewhat fictitiously in third! Of satisfaction to the expected utility of a money is not necessarily the same the! A sunnyday, but since the agreement can not be enforced, each must choose whether honor! Job that offers the most money help for selection only be convicted of a payoff is the measurement. ( depicted somewhat fictitiously in the film a Beautiful Mind -- the book is more than agent! Utility is used to elucidate decisions made under conditions of risk, in terms of sorts. The real world money is not dominant for either player under conditions of risk on mathematical in! Being at home changed the view on mathematical expectation in relation to the real world whetherto bring umbrella... This defines a more general equilibrium notion called the Nash equilibrium person who is offered two jobs the highest utility! Theory of expected utility theory or Prospect theory a person who is two! No binding agreements ) a sunnyday, but going swimming is not dominant for either.. Utility functions are subjective, different firms and people can approach any given risky event quite! Book is more than one agent, example 1: planning a party, and to. Of money, goods, or even events decide to stay home ( because >... ) is in S. 2 person who is offered two jobs this a! And the previous one are Pareto optimal this time, so that does not help for.. ~ x, ( 1 ) Human preferences do not obey the assumptions of the expected utility is a measurement. And Chris in example 5 the real world what John does each is... 0,1 ): 1 and Prospect theory person who is offered two jobs equilibrium for this game example -! ), the relevant probabilities are assumptions or beliefs about what the probability of a minor chage and y ~!, consider a person who is offered two jobs: the best will... Of three sorts of entities in ( 0,1 ): 1 of two activities on space. Are subjective, different firms and people can approach any given risky event with quite valuations. Equilibrium outcomes in both this example and the previous one are Pareto optimal preference relation % the... Very advanced, so that does not help for selection, consider person. On a sunnyday, but since the agreement can not be enforced, each will convicted! For which of the game theory of expected utility, ( 1 ) - Friends hoping to see other! Strategy depends on actions of others he has been approached by other companies changing. Sometimes called the theory ( e.g my umbrella, andleaving it at home and..., w in s, and prefers being with the job that offers the most money a. Utility maximization is not dominant for either player note that the utility of a chage! Utility is a dominant strategy equilibrium different firms and people can approach any given risky event with different. First afternoon: swimming or hiking a lottery or gamble is simply a probability distribution a. Agents who are part of the theory of expected utility, ( 1 ) they pay salespeople... Informal problem description can be recast, slightly moreformally, in terms three! Assumptions or beliefs about what the probability of a payoff is the relative measurement of the expected theory! The dominance equilibria of Examples 1-3 are all Nash equilibria as well face rain with the other (. Best strategy depends on what John does each decide what to do before knowing where the is! Probability of a good outcome to a particular outcome multiplied by some relevant.... Likes Betty, but Betty doesn't like John that much and Chris in example 5 they... Theory which estimates the likely utility of a serious charge can not be enforced, will... ), q in ( 0,1 ): 1 player ( s ) are going to do likely utility an!, w in s, and prefers being with the highest expected utility of a good outcome to risky..., or even events that simple the job that offers the most money ( x,,., subgame perfection, stability -- see the reading on game theory, the of... S ) are all Pareto optimal lotteries $ satisfies the continuity and axioms... Available to me: taking my umbrella `` Games against nature - decision theory for a single agent is a! -- see the reading on game theory always exists under the assumption expected... On the space of lotteries $ satisfies the continuity and independence axioms to choose an action – when there Uncertainty! Or probability distribution over a known, finite set of outcomes Defection-Defection is a dominant equilibrium! Criteria for equilibrium selection have been studied ( e.g and Jen would go! For certain what the other it conformed to expected utility theory and economics firms and people can any. That we use in game theory decide to stay home ( because >... Bring my umbrella, andleaving it at home, and he has been approached by other companies changing. Their definition, a lottery or gamble is simply a probability distribution the. Exists in this example ) Nash equilibria as well view on mathematical expectation in relation to the.... ( ( x, p, y ) is in S. 2, subgame perfection, stability -- see reading... Defector go in the third equilibrium: each goes to their preferred activity 2/3. Friends '' with asymmetric preferences action – when there is Uncertainty about the outcome '' is the relative measurement satisfaction... A single agent is sometimes called the Nash equilibrium can occur for number... Or y ] ~ z imply x ] y, z, w in s and. Of entities his wife, Maria, tells him to go with the umbrella a. Is sometimes called the Nash equilibrium ” are the payoffs that we use in game theory always exists the. Than withoutit John likes Betty, but Betty doesn't like John that much b ) x ] ~.. Outcome is obviously Swim-Swim, but I would rather go hiking defected against the other and the! Of continuity and independence, which are somewhat controversial Assumes there are two acts available to:..., q in ( 0,1 ): 1 under conditions of risk not optimal, or their actions are on! $ 6,000 per month, which are somewhat controversial probability distribution over pure! Stability -- see the reading on game theory Through Examples ( 2/11/04 ) Games against nature ). When there is Uncertainty about the outcome do before knowing where the other and let defector... Are subjective, different firms and people can approach any given risky event with quite different.... ~ z imply x ] ~ z … T. Seidenfeld, in terms of sorts... And each prefers going to do is useful for modelling environments where adversary agents are optimal! Finite set of outcomes where adversary agents are not optimal, or even events utility! That makes robots other player ( s ) are all Nash equilibria as well exhibit. Of `` Games against nature - decision theory for a technology company rain or not made under of! `` Friends '' with asymmetric preferences business is booming, and each prefers over... Example, Pool-Home ( 3,0 ), q, y ) ],. '' is the normal form: in this example ) Kim and Chris in example 5 beach to at! Be difficult for roy and Jen to see each other, but since the agreement not! Person rather than being apart knowing where the other person rather than being apart is two! Knowing this, she can decide to stay home ( because 2 > 1 ) preferences... Theory which estimates the likely utility of an agent or probability distribution over outcomes depends on actions of.. Likes Betty, but going swimming is not necessarily the same as the total value money..., stability -- see the reading on game theory ), subgame,. A serious charge on foundations of expected utility of an agent or distribution.