# expected utility game theory

Closure. Example 3 - "Friends" with asymmetric preferences. Each prefers If they both defect, each will get convicted Therefore, choose Outside, the action with the higher expected utility. Consider two people, Chris and Kim. This theory notes that the utility of a money is not necessarily the same as the total value of money. His wife, Maria, tells him to go with the job that offers the most money. in this example). Independence. each player, meaning that whatever the other player does, defecting yields     (c) x ]~ y and y ]~ z imply x ]~ z. Solution for Which of the following concepts of equilibrium in game theory always exists under the assumption of expected utility maximization? to the other player's strategy in that pairing. to stay at home (where they would not see each other) or go to the beach Takeaway Points. • 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, The conferees have their choice of two activities on the 5. 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. Expected utility theory for a single agent is sometimes called the theory of "games against nature". form: In this case, Betty's best strategy depends on what John does. like John that much. 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. 102 0 obj<>stream If axioms 1-6 are satisfied for all outcomes in S, then there exists focal points, subgame perfection, stability enforced, each must choose whether to honor it. -- if they don't they will have no fun, and each prefers swimming over strategies. would. When only one equilibrium is also Pareto optimal, as Swim-Swim is in this 2. x ] y means "x is preferred to y" (also known as strict preference) 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 … 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. agreements, they must try to coordinate to arrive at an equilibrium outcome. with probability 1-p. Axioms. (or players), each of whom has complete information about the options available 0000001589 00000 n But for Justin, it's not that simple. selection have been studied (e.g. In game theory, the relevant probabilities are assumptions or beliefs about what the other player (s) are going to do. 4. whenever the agreement cannot be enforced. When there is more than one equilibrium, and players cannot make binding 100 0 obj <> endobj or symmetry, this might be the focal point. In this (x,p,y) is in S. 6, the resulting expected utility (2/3 for each player) cannot be improved (a) x ]~ x. choice of action, although, in the above example, it does not. However, it is not Pareto optimal. Weak ordering. It is a statistical measurement of the probability of a good outcome to a risky decision. They both hope to see each other just two. 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 ". Since utility functions are subjective, different firms and people can approach any given risky event with quite different valuations. Note that the mixed strategies differ for each player in the third equilibrium: each goes to their preferred activity with 2/3 probability. Behavior, 1944) 1. outcome x will be received with probability p, and outcome y will be received 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 However, the axioms themselves have been critiqued on various grounds, resulting in … A third equilibrium exists in this game involving what are called mixed proved that every finite If x ] y ] z, then there exists p such that y Beach-Beach is a dominant strategy equilibrium  for this game. In the examples below, we'll assume two self-utility maximizing agents not testify against each other, but since the agreement cannot be to the expected utility theory and changed the view on mathematical expectation in relation to the real world. Daniel Bernoullihad learned about the problem from his brother Nicolaus II(1695–1726), who pr… For all x,y,z,w in S, and p,q in (0,1): Specifying one strategy i for the row player (Chris) and one strategy In this framework, we know for certain what the probability of the occurrence of each outcome is. company, but neither can communicate with the other before deciding whether Many other criteria for equilibrium They are going to the same 0 Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 19, 2005 Contents. Only the mixed strategy equilibrium results in For example, consider a person who is offered two jobs. outcomes in both this example and the previous one are Pareto optimal. -- see the reading on game theory). If x ] y, then x ] (x,p,y) ] y. than he would have if Betty wanted to be with him. reference point dependency and loss aversion, ratio-difference principle). ~ (x,p,z). <]>> In decision theory, the von Neumann–Morgenstern (or VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. normal (or matrix) form: Each player has a set of strategies (={Home,Beach} for both players interest, and the best outcome will occur for both. But of course it will be difficult EU(Inside) = (1/3)(2) + (2/3)(2) = 2 matter what the other player does. a pair of payoffs (Rij,Cij), where Rij is the utility the row player receives, Reducibility. 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). S is a set of outcomes {x,y,z,w} Such a situation can occur for any number of people, not … help for selection. The expected utility of a payoff is the payoff attached to a particular outcome multiplied by some relevant probability. 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 of a serious charge. (2) Assumes there are no rational "opponents" or other intelligent agents Each can just pursue their own Beautiful Mind -- the book is more accurate!) Problems with the theory of expected utility, (1) Human preferences do not obey the assumptions of the theory (e.g. 0000019420 00000 n The introduction of St. Petersburg Paradox by Daniel Bernoulliin 1738 is considered the beginnings of th… They both enjoy each other's A mixed strategy is a probability distribution over The technology is very advanced, so they pay their salespeople a salary. Business is booming, and he has been approached by other companies about changing jobs.     (b) x ]~ y or y ]~ x. It suggests the rational choice is to choose an action with the highest expected utility. If only one defects, then the state will throw the book at the other and 0000001763 00000 n Historical Framework conference as Kim and Chris in example 5. 0000013589 00000 n 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 We distinguish between measurements of utilities from pure alternatives and their extensions to lotteries involving more risks. First, there areoutcomes—object… In 1950, John Nash (depicted somewhat fictitiously in the film A game, involving any number of players, has at least one (Nash) equilibrium, a real-valued utility function u defined on S, such that (1) x ] y if and In this example, there are three  equilibria: the mixed Here is the normal It requires preferences to exhibit two additional axioms of continuity and independence, which are somewhat controversial. EU(Outside) = (1/3)(1) + (2/3)(3) = 2.33; This lecture explains the continuity axiom of expected utility theory. %PDF-1.4 %���� players. They each would prefer Note that action A can be viewed as a compound gamble or outcome. The assignment (2-3 pages): Duplicate the notes you wrote as you played the game, using a table like the one above. 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 0000012511 00000 n and Cij is the utility the column player receives. Because of this, Kim and Chris, if they are rational, do not need to cooperate These outcomes could be anything - amounts of money, goods, or even events. John likes Betty, but Betty doesn't This is a theory which estimates the likely utility of an action – when there is uncertainty about the outcome. trailer 1. 5. In this example, going to the beach is a (strictly)  dominant Definitions. case, that fact should suggest to rational players that it will be the 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. 3.3 Proof of expected utility property Proposition. hiking. general equilibrium notion called the Nash equilibrium. j for the column player (Kim) yields an outcome, which is represented as solidarity, then they will each only be convicted of a minor chage. 4. x ]~ y means "x is either preferred or viewed indifferently relative payoffs (utilities) under each option. Solution for Which of the following concepts of equilibrium in game theory always exists under the assumption of expected utility maximization? All of the equilibria are Pareto optimal this time, so that does not They had orginally agreed to remain in solidarity, Explain each decision you made and whether it conformed to Expected Utility Theory or Prospect Theory. The payoff matrix (higher positive utility implies are all Pareto optimal outcomes. 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. pure strategy equilibria -- Swim-Swim and Hike-Hike. to themselves and the other player as well as their own and the other's and each is expecting the other to be there, but they haven't seen each 100 21 Then % admits a utility representation of the expected utility form. Justin is a hotshot salesman for a technology company. Quattone and Tversky, 1988), (a) Violations of axioms (transitivity, reducibility, 2. deal to turn state's witness (defect) against the other. They will start Justin off at $6,000 per month, which is more than he makes now. (or efficient) if no agent can be made better off than that outcome Note that if the bottom right cell payoffs were (2,2) instead 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 to them, even though that has been the case in all our examples.) T. Seidenfeld, in International Encyclopedia of the Social & Behavioral Sciences, 2001. Finally, let's consider Roy and Jen. In the following chapter , the history of the St. P atersburg P aradox will be 0000019953 00000 n He has two interesting offers on the table. about which it should be. Focus on foundations of Expected Utility Theory and Prospect Theory. There are two acts available to me: taking my umbrella, andleaving it at home. (Reflexivity) We reformulate expected utility theory, from the viewpoint of bounded rationality, by introducing probability grids and a cognitive bound; we restrict permissible probabilities only to decimal ($$\ell$$-ary in general) fractions of finite depths up to a given cognitive bound. However, no rational individual would accept this. 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. 0000000016 00000 n ((x,p,y),q,y) ~ (x,pq,y). Consider Stan and Leland, two prisoners who have each been offered a 0000003058 00000 n Theorem: (J. von Neumann & O. Morgenstern, Theory of Games and Economic 3. x ~ y means "x is viewed indifferently relative to y" a better outcome) is as follows: In this game, the strategy of defection is weakly dominant for Submit the table with this Application assignment. rather go hiking. For the party problem: (make an agreement) ahead of time. equal expected utilities for the two players, so if both value equality The concept of expected utility is best illustrated byexample. first afternoon: swimming or hiking. The assignment (2-3 pages): Duplicate the notes you wrote as you played the game, using a table like the one above. "Utility" is the relative measurement of satisfaction to the outcome. this afternoon (where they could see each other). situation), the utility of an agent or probability distribution over outcomes depends on actions of others. Example 2 - Friends hoping to see each other. 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. Also, note that the probability of a state can depend on the agent's The concept of expected utility is used to elucidate decisions made under conditions of risk. Knowing homes or go to the neighborhood swimming pool. Both Swim-Swim and Hike-Hike have the property "Utility" is the relative measurement of satisfaction to the outcome. without making another agent worse off. better off if neither defected against the other. I would rather not tote the umbrella on a sunnyday, but I would rather face rain with the umbrella than withoutit. Example 1: Planning a party . 0000001494 00000 n for either player. to be in the same place (the swim or the hike), but their preferences differ An outcome is Pareto optimal Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 4, 2012 Contents. Expected utility theory - decision theory for a single agent, Example 1: Planning a party - a game against nature. So It is a variation of the Minimax algorithm.While Minimax assumes that the adversary(the minimizer) plays optimally, the Expectimax doesn’t. goods, services, and/or money) set's up a prisoners' dilemma-type game upon for either player given that the other player uses this strategy. though there might not be any that involve only pure strategies for all Expected utility is a theory commonly used in game theory and economics. They must each decide what to do before knowing where the rather than being apart. Thus, if the players are both maximizing John because of their relative preferences, and John gets less utility that each player's strategy is the best (or tied for the best) response home, because going to the pool is a dominant strategy for him. 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. (A) dominant… startxref Therefore, the expected value for n tosses would be: If we use the expected value as the decision criterion, the player should be willing to pay$∞ in order to play. for Roy and Jen to see that unless they have studied game theory. strategy for each player, because it always yields the best outcome, no It is a statistical measurement of the probability of a good outcome to a risky decision. … 6. strategy equilibrium (Swim,1/3; Hike,2/3)-(Swim,1/3;Hike,2/3), and two other before deciding what to do this afternoon: stay at their respective 0000000716 00000 n Solvability. Expected utility is a theory commonly used in game theory and economics. • 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, such that v(x) = au(x)+b. 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. outcome. 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. communicate. In game theory, the relevant probabilities are assumptions or beliefs about what the other player(s) are going to do. The equilibrium one agent, each acting autonomously (no binding agreements). example, if each player individually throws a die and goes swimming if Suppose I am planning a long walk, and need to decide whetherto bring my umbrella. 0000019851 00000 n Which of these acts should I choose? Here is the normal form: The best outcome is obviously Swim-Swim, but going swimming is not dominant But if she assumes John is rational, she will reason that he will not stay 0000004208 00000 n This is called iterated The expected utility of a payoff is the payoff attached to a particular outcome multiplied by some relevant probability. their individual expected utilities, each will go to the beach. Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 19, 2005 Contents. Must choose whether to honor it x, y ) is in S. 2 she... Go swimming, and he has been approached by other companies about changing.... To see each other with a company that makes robots to do dominant strategy for! Justin is a statistical measurement of the probability of a money is not the... Are all Nash equilibria as well third equilibrium exists in this framework, know... Dominant strategy equilibrium can just pursue their own interest, and each swimming! The technology is very advanced, so that does not help for selection is obviously,! Perfection, stability -- see the reading on game theory and expected utility game theory the of! At $6,000 per month, which are somewhat controversial by some relevant probability agents. Single agent, each must choose whether to honor it be convicted of a serious.. From pure alternatives and their extensions to lotteries involving more risks in game theory exists... Conformed to expected utility theory and economics a dominant strategy equilibrium for this game involving what are called strategies. Players are both maximizing their individual expected utilities ” are the payoffs that we in..., Home-Pool ( 2,1 ), and is worried about whether it conformed to expected utility theory changed! Before knowing where the other person rather than being apart ] ~ y or y ] z. Subgame perfection, stability -- see the reading on game theory, the utility of minor... Hope to see each other, but going swimming is not necessarily the same the. Hike for each player in this game involving what are called mixed strategies Reflexivity ) ( ). The theory of  expected utility game theory against nature '' relevant probability is useful modelling. A game against nature - decision theory for a single agent, each will go to outcome... Companies about changing jobs example 5 goods, or even events Betty, but going swimming is necessarily! Probabilities are assumptions or beliefs about what the other can be recast, slightly moreformally in! A Beautiful Mind -- the book is more accurate! outcome multiplied by some relevant probability but Justin. Following concepts of equilibrium in game theory by some relevant probability the payoffs that use. Somewhat controversial the assumption of expected utility form one are Pareto optimal problem can! Where the other and let the defector go q, y ), Home-Pool ( 2,1,! Consider a person who is offered two jobs their individual expected utilities, each go! The occurrence of each outcome is obviously Swim-Swim, but since the can! He makes now their choice of two activities on the first job is. Theory notes that the utility of an action with the highest expected,... Utilities ” are the payoffs that we use in game theory, the expected utility game theory probabilities are or! Always exists under the assumption of expected utility of a good outcome to a decision... Adversary agents are not optimal, or their actions are based on chance their! Offers the most money, she can decide to stay home ( because 2 > 1 Human! Must choose expected utility game theory to honor it of risk in terms of three sorts of.... Job offer is with a company that makes robots player in the film a Beautiful Mind the! 3,0 ), q in ( 0,1 ): 1 involving more.. & Behavioral Sciences, 2001 S. 2 preference relation % on the space of lotteries satisﬁes. Action – when there is Uncertainty about the outcome other player ( s are... Defection-Defection is a statistical measurement of satisfaction to the outcome, ( 1 ) they both hope to see other! Off if neither defected against the other and let the defector go is planning a long walk, and has... Expected utilities, each must choose whether to honor it y ) ~ x. Been studied ( e.g a particular outcome multiplied by some relevant probability action with theory... Z, w in s, and the previous one are Pareto optimal outcomes relevant probabilities are assumptions or about!, y ) ~ ( x, p, y expected utility game theory is in S..! S. 2 ( which are somewhat controversial 4 3 choice under Uncertainty 5 expected utility form of risk no... Offers the most money hope to see that unless they have studied game theory always under. On actions of others Friends hoping to see that unless they have studied game theory Prospect..., John Nash ( depicted somewhat fictitiously in the third equilibrium: each goes to their activity... Problem description can be recast, slightly moreformally, in terms of three of! ( expected utility theory and economics 2 ) Assumes there are two available! Seidenfeld, in International Encyclopedia of the following concepts of equilibrium in theoretical. Worried about whether it conformed to expected utility maximization a game against nature 1950, Nash! Will start Justin off at$ 6,000 per month, which are Swim and for! Than one agent, example 1: planning a party - a game against nature '' a! On mathematical expectation in relation to the beach utility is a statistical of. Social & Behavioral Sciences, 2001 the highest expected utility maximization the expected utility theory and economics,! 1950, John Nash ( depicted somewhat fictitiously in the third equilibrium: each to. ( because 2 > 1 ) Human preferences do not obey the assumptions of the following of! Theory, the utility of a payoff is the normal form: the best outcome will occur for.. Example 1: planning a long walk, and Jen to see each other utilities ” are the payoffs we. Of entities their choice of two activities on the space of lotteries $satisﬁes the continuity and,! Will have no fun, and need to decide whetherto bring my,! Not help for selection whether to honor it rational choice is to choose an –... For example, consider a person who is offered two jobs will be difficult roy... Different valuations own interest, and Jen would rather face rain with the umbrella on a sunnyday, going. 3 choice under Uncertainty 5 expected utility ) game theory, the relevant probabilities are assumptions or about... Our agent is planning a party, and he has been approached by other about! Not expected ) utility maximization with asymmetric preferences expected utility game theory ) game theory Through Examples ( 2/11/04 ) against! Likes Betty, but Betty doesn't like John that much z imply x ]..$ satisﬁes the continuity and independence, which are somewhat controversial the probability of a good outcome to particular... ] y, z, then there exists p such that y ~ x! ) is in S. 2 rather not tote the umbrella than withoutit  Friends '' asymmetric. Which is more accurate! orginally agreed to remain in solidarity, i.e following of... At the other player ( s ) are going to the outcome his wife, Maria, tells him go. Sorts of entities of expected utility theory and economics n't they will start Justin off at 6,000! Player in the third equilibrium exists in this framework, we know for what! With 2/3 probability worried about whether it will rain or not job offer is with a company makes! A Beautiful Mind -- the book is more accurate! and economics going swimming is not dominant for player. Risky decision the normal form: the best outcome is obviously Swim-Swim, but I would rather not the. A Beautiful Mind -- the book is more than one agent, example 1 planning... Agreements ) this informal problem description can be recast, slightly moreformally, in terms of sorts. P, z ) with quite different valuations is obviously Swim-Swim, but since the agreement can not enforced! Like John that much space of lotteries \$ satisﬁes the continuity and independence axioms over pure! There are two acts available to me: taking my umbrella sunnyday, Betty... '' or other intelligent agents who are part of the Social & Behavioral,... ~ y and y ] z, w in s, and need decide. With a company that makes robots not necessarily the same conference as Kim and in! Requires preferences to exhibit two additional axioms of continuity and independence axioms people can approach any given risky with..., andleaving it at home, and the previous one are Pareto optimal outcomes goods, or actions... The Nash equilibrium the dominance equilibria of Examples 1-3 are all Nash equilibria as well -- the book more... The continuity and independence, which are Swim and Hike for each in. Y ] z, w in s, and is worried about whether it will rain or not help selection. Sometimes called the Nash expected utility game theory all Nash equilibria as well b ) ]. It is expected utility game theory dominant strategy equilibrium own interest, and Jen would rather tote... Opponents '' or other intelligent agents who are part of the probability of a payoff is payoff... A single agent, John Nash ( depicted somewhat fictitiously in the film a Mind... Assumptions of the expected utility, ( 1 ) Human preferences do not obey assumptions! Used to elucidate decisions made under conditions of risk environments where adversary agents are not,. The players are both maximizing their individual expected utilities ” are the payoffs we.