Introduction: Noncooperative Games Dominant and …hrtdmrt2/Teaching/GT_2017_19/L1.pdfA Brief...

Introduction: Noncooperative Games Dominantand Dominated Strategies

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

Jun 20th, 2017

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda1 What do we do in Economic?2 What is Game Theory?3 A Brief History of Game Theory4 The Theory of Rational Choice5 The Extensive Form Representation of a Game6 Strategies and the Normal Form Representation of a Game7 The Rational Choice Paradigm8 Exercises9 Formalizing the Game

10 Dominant and Dominated Strategies11 Iterated Delation of Strictly Dominated Strategies12 Iterated Delation of Dominated Strategies13 Exercises


What do we do in Economic?






I What makes a theoretical model ”economics” is that the concepts we areanalyzing are taken from real life.

I Through the investigation of these concepts, we indeed try to understand realitybetter, and the models provide a language that enables us to think abouteconomic interactions in a systematic way.

I We do not view economic models as an attempt to describe exactly the world, orto provide tools for predicting the future.

I Although we will be studying formal concepts and models, they will always begiven an interpretation. An economic model differs substantially from a purelymathematical model in that it is a combination of a mathematical model and itsinterpretation.

C. Hurtado (UIUC - Economics) Game Theory 1 / 66



I The word ”model” sounds more scientific than ”fable” or ”fairy tale”, but there isnot much difference between them. The author of a fable draws a parallel to asituation in real life and has some moral he wishes to impart to the reader.

I The fable is an imaginary situation that is somewhere between fantasy and reality.Any fable can be dismissed as being unrealistic or simplistic, but this is also thefable’s advantage.

I Being something between fantasy and reality, a fable is free of extraneous detailsand annoying diversions. In this unencumbered state, we can clearly discern whatcannot always be seen from the real world.

I On our return to reality, we are in possession of some sound advice or a relevantargument that can be used in the real world. We do exactly the same thing ineconomic theory.

I Thus, a good model in economic theory, like a good fable, identifies a number ofthemes and elucidates them. We perform thought exercises that are only looselyconnected to reality and have been stripped of most of their real-life characteristics.

I However, in a good model, as in a good fable, something significant remains.



As if Rationality

I Rationality forms the basis of decision-making in the neoclassical school.

I Decision-makers are optimizers, given the constraints they find themselves in.

I Rationality assumes that decision-makers maximize things that give themhappiness and minimize things that give them pain.

I Implications:

- Narrows down the set of possible outcomes.

- Rational man is a clever individual.

- Rationality helps to predict the outcome of an economic system.

- Once economic agents have optimized their utility and reached a situationwhere they do not want deviate, the economic system reaches a stableoutcome: ’equilibrium’.



Perfect Competition is a Benchmark

I Assumes the existence of many buyers and many sellers.

I Decision making is made independently and individually.

- Decisions are not made in coalitions (together/jointly)

- Decisions are not inter-dependent: Decision of one agent is neitherinfluenced by another agent, nor does it influence that of another agent.

I Independent and individual decision-making under perfect competition implies eachdecision-maker tries to do the best they can irrespective of what otherdecision-makers are doing. (really?)

I Perfect competition is a theoretical extreme. Like the ideal human bodytemperature of 98.4 degrees Fahrenheit it almost never exists. It is used as abenchmark to explain deviations from this ’perfect’ world.


What is Game Theory?






I Branch of applied mathematics and economics that studies strategic situationswhere there are several players, with different goals, whose actions can affect oneanother.

I A game is any situation where multiple players can affect the outcome, a player isa stakeholder, a move or option is an action a player can take and, at the end ofthe game, the payoff for each player is the outcome.

I The value of game theory lies in understanding the interactions and likelyoutcomes when the end result is dependent on the actions of others who havepotentially conflicting motives.

I Game theory is mainly used in economics, political science, and psychology, as wellas logic, computer science, and biology.

I The subject first addressed zero-sum games, such that one person’s gains exactlyequal net losses of the other participant or participants. Today, game theoryapplies to a wide range of behavioral relations: the study of decision science,including both humans and non-humans.




Let’s play a game!

I Each person writes a number between 0 and 100.

I The winner is the one who wrote the closest number to half the average of allnumbers, including his or her own number.


A Brief History of Game Theory






1713In a letter dated 13 November 1713 Francis Waldegrave provided the first known,minimax mixed strategy solution to a two-person game. Waldegrave wrote theletter, about a two-person version of a card game, to Pierre-Remond de Montmortwho in turn wrote to Nicolas Bernoulli, including in his letter a discussion of theWaldegrave solution.

1838The French economist Antoine Augustine Cournot discussed a duopoly where thetwo duopolists set their output based on residual demand.

1871In the first edition of his book The Descent of Man, and Selection in Relation toSex Charles Darwin gives the first (implicitly) game theoretic argument inevolutionary biology: If births of females are less common than males, females canexpect to have more offspring. Thus parents genetically disposed to producefemales tend to have more than the average numbers of grandchildren, hence,female births become more common. As the 1:1 sex ratio is approached, theadvantage associated with producing females dies away.




Intellectual debate between the French Mathematician Emile Borel (1871-1956)and the Hungarian Mathematician John Von Neumann (1903-1957):

Do zero-sum games have a solution?

1921Emile Borel published four notes on strategic games and an erratum to one ofthem. Borel gave the first modern formulation of a mixed strategy along withfinding the minimax solution for two-person games with three or five possiblestrategies. Initially he maintained that games with more possible strategies wouldnot have minimax solutions, but by 1927, he considered this an open question ashe had been unable to find a counterexample.

1928John von Neumann proved the minimax theorem. It states that every two-personzero-sum game with finitely many pure strategies for each player is determined, ie:when mixed strategies are admitted, this variety of game has precisely oneindividually rational payoff vector.




1934R.A. Fisher independently discovers Waldegrave’s solution to the card game. Fisherreported his work in the paper Randomisation and an Old Enigma of Card Play.

Intellectual debate between Hungarian Mathematician John Von Neumann(1903-1957) and he Austrian economist Oskar Morgenstern (1902-1977):

Can utility be quantified?

1944Theory of Games and Economic Behavior by John von Neumann and OskarMorgenstern is published. As well as expounding two-person zero sum theory thisbook is the seminal work in areas of game theory such as the notion of acooperative game, with transferable utility (TU), its coalitional form and its vonNeumann-Morgenstern stable sets. It was also the account of axiomatic utilitytheory given here that led to its wide spread adoption within economics.




Von Neumann and Morgenstern formally laid the foundations of Game Theory as abranch of applied mathematics. However, their effort failed to generate stir for tworeasons.

1. The Theory of Games and Economic Behavior was based on the notion ofzero-sum games. These are known as ’Games of Conflict’ or ’Non-CooperativeGames’. Most games in social sciences are non-zero-sum games.

2. It did not establish how equilibrium in games of interdependent decision-makingwould arise. The world did not have to wait too long for this solution.




1950Contributions to the Theory of Games I, H. W. Kuhn and A. W. Tucker eds.,published.

1950In January 1950 Melvin Dresher and Merrill Flood identified a game where it is inthe best interest for players to cooperate, but individual self-interest invokes themto not cooperate. The game reaches a bad equilibrium that is inferior to a superioroutcome that could have been reached and was available. This phenomenon wascanonized by Albert Tucker.

In the summer of 1950 Tucker was at Stanford University. He was working on aproblem in his room when a graduate student of psychology knocked and askedwhat he was doing. The answer was short: game theory. ”Why don’t you explainto us in a seminar”? Tucker used his now famous example of two thieves who wereput into separate cells and asked the same question by the judge. Tuckerchristened the phenomenon as The Prisoners’ Dilemma.




1953Extensive form games allow the modeler to specify the exact order in which playershave to make their decisions and to formulate the assumptions about theinformation possessed by the players in all stages of the game. In two papers,Extensive Games (1950) and Extensive Games and the Problem of Information(1953), H. W. Kuhn included the formulation of extensive form games which iscurrently used, and also some basic theorems pertaining to this class of games.

1953In four papers between 1950 and 1953 John Nash (1928 - 2015) made seminalcontributions to both non-cooperative game theory and to bargaining theory.




Note: Nash arrived at Princeton in the Fall of 1948 to start a PhD. He came toPrinceton with one of the shortest reference letters. His professor Richard Duffin(1909-1996) wrote just one sentence: ”He is a mathematical genius.”

Just eighteen months later, Nash submitted a 28 page doctoral dissertation. Thenumber 28 went on to become a superstitious number at Princeton. VonNeumann solved the mini-max theorem in 1928. Nash was born in 1928. Nash’sdoctoral dissertation was 28 pages.



A Brief History of Game Theory1953

In two papers, Equilibrium Points in N-Person Games (1950) and Non-cooperativeGames (1951), Nash proved the existence of a strategic equilibrium fornon-cooperative games -the Nash equilibrium- and proposed the ”Nash program”,in which he suggested approaching the study of cooperative games via theirreduction to non-cooperative form.

- This was the missing link to Von Neumann and Morgenstern’s magnum opus!!- He established equilibrium where all players calculate their best strategy based on

what they assume is the best strategy of the other players.- Every finite game must have at least one solution such that once reached, no

player within the game will have an incentive to deviate.- If all rational economic agents in a system are trying to do the best they can, the

economic system must be in equilibrium such that no single agent will want tounilaterally deviate from their position.

- A Nash equilibrium does not necessarily imply that a game will reach a solutionthat is the best possible solution for all players in the game.




Why was John Nash’s 1950 Game Theory paper such a big deal?

I In 1928, Hungarian mathematician John von Neumann, launched the field of gametheory with a study of two-player ”zero-sum” games, in which the players havecompletely opposite interests

I von Neumann dealt with the case n = 2, and it was by no means obvious how toextend the concept of equilibrium for the general case and prove that it alwaysexists.

I Nash is very clear about this in his 1951 Annals paper:The notion of an equilibrium point is the basic ingredient in our theory. This

notion yields a generalization of the concept of the solution of a two-personzero-sum game. It turns out that the set of equilibrium points of a two-personzero-sum game is simply the set of all pairs of opposing ”good strategies.”

I The solution concept was very innovative. For example, Arrow and Debrew (1954)generalized Nash’s theorem on the existence of equilibrium points for games toproof the existence of an equilibrium for a competitive economy.




1953The notion of the Core as a general solution concept was developed by L. S.Shapley (Rand Corporation research memorandum, Notes on the N-Person GameIII: Some Variants of the von-Neumann-Morgenstern Definition of Solution, RM-817, 1952) and D. B. Gillies (Some Theorems on N-Person Games, Ph.D. thesis,Department of Mathematics, Princeton University, 1953). The core is the set ofallocations that cannot be improved upon by any coalition.

50’sNear the end of this decade came the first studies of repeated games. The mainresult to appear at this time was the Folk Theorem. This states that theequilibrium outcomes in an infinitely repeated game coincide with the feasible andstrongly individually rational outcomes of the one-shot game on which it is based.Authorship of the theorem is obscure.




I Identification and formalization of the Prisoners’ Dilemma by Flood, Dresher andTucker had important implications in economic theory and the social sciences.

- It provided a rigorous explanation to why intervention into markets can bejustified in the presence of public goods that are jointly consumed onceprovided, but almost impossible to finance through voluntary contribution.

I This phenomenon formed the basis to why players in a game find a mutual interestto cooperate if a Prisoners’ Dilemma game is played more than once. This lead towhat is known as Trigger Strategies.

- Anatol Rapoport (1911-2007) proposed a simple strategy called Tit For Tatin a repeated Prisoners’ Dilemma tournament.

- Rapoport proposed, each player cooperates with the other player in arepeated Prisoners’ Dilemma as long as the other player does the same. Ifone player defects in one round, the game ends with the other playerapplying the Trigger of Tit For Tat by no cooperation and ending the game.




I Refinement of the Nash Equilibrium: The development of Game Theory from the1950s was based on the limitations of the Nash Equilibrium.

- The Nash Equilibrium argued that a game can have multiple equilibria(solutions). It did not provide how to identify the set of multiple equilibriaand which one to choose as the more probable equilibrium.

I Thomas Schelling introduced two terms Focal Point (the logical outcome frominformation from outside a game) and Credible Commitment (sending a signal thata player will commit to a certain actions).

- The Focal Point provided a basis for behavioural sciences why players preferone set of equilibrium to another. It also helped narrow down probablesolutions from a larger set.

- The Credible Commitment (Threat) gave an added edge to explain previousmodels like the Stackelberg model in a sequential game setting.




I A further development in the Nash Equilibrium came through the works of theHungarian-American Game Theorist John Harsanyi.

I The Nash equilibrium was based on games of complete information. In economictheory, this is known as games under symmetric information when all players in agame make decisions based on having the same set of information.

I By the 1970s, economists like George Akerlof, Michael Spence and Joseph Stiglitzstarted analyzing decision-making under asymmetric information or games underincomplete information.

I Harsanyi had by that time refined the Nash Equilibrium under BayesianProbability. This enabled the analysis of games where different players havedifferent sets of information about themselves.

I Harsanyi’s refinement was a blessing for economists who were battling withasymmetric information.




Goodbye to Rationality:

I By the 1960s and the 1970s, psychologists studying decision-making, started tochallenge the rationality assumption upon which the neoclassical school ofeconomic theory and Game Theory was based.

I One of the pioneers was Herbert Simon (1916- 2001). He introduced the notion ofBounded Rationality.

I When individuals make decisions they posses limited information, limited cognitiveability, and limited time within which to make decisions.

I With the development of bounded rationality, economic theory slowly started tobecome a behavioural science.




I Robert Rosenthal (b 1933) first introduced the Centipede Game in 1981.

I Two players take turns choosing either to take a slightly larger share of a slowlyincreasing pot, or to pass the pot to the other player. Based on sub-game perfectequilibrium and backward induction, the rational outcome of the game would befor the player making the first decision to take the entire pot in the first round.

I However, empirical evidence suggested otherwise. Players tend to partiallycooperate so the pot becomes larger as the game proceeds. This empiricaloutcome challenged the rationality assumption of Game Theory.




I The Ultimatum Game was introduced in 1982 by Werner Guth, RolfSchmittberger, and Bernd Schwarze.

I In the Ultimatum Game, two players decide how to divide a sum of money. Thefirst player proposes how to divide the sum between the two players. The secondplayer can either accept or reject this proposal. If the second player rejects, neitherplayer receives anything. If the second player accepts, the money is split accordingto the proposal. The game is played only once.

I This game challenges the rationality assumption. Rationality would suggest theproposer keeps all the money and does not offer anything to the other player.

I Empirical evidence suggests ethics and morality and evolutionary behaviour leadsto some degree of fairness in the splitting of the money.

I Extension of this is the Rubinstein Bargaining: Two players that alternate theultimatum game. The game keeps going until one player accepts an offer.Rubinstein uses discount factors (every delay is costly). Rubinstein recovers aunique solution.


The Theory of Rational Choice






I A decision-maker chooses the best action according to her preferences, among allthe actions available to her.

I Her ”rationality” lies in the consistency of her decisions when faced with differentsets of available actions, not in the nature of her likes and dislikes.

Actions:- A set A consisting of all the actions that, under some circumstances, are

available to the decision-maker.- In any given situation the decision-maker is faced with a subset of A, from

which she must choose a single element.- The decision-maker knows this subset of available choices, and takes it as

given; in particular, the subset is not influenced by the decision-maker’spreferences.

- The set A could, for example, be the set of bundles of goods that thedecision-maker can possibly consume; given her income at any time, she isrestricted to choose from the subset of A containing the bundles she canafford.




Preferences and payoff functions:- We assume that the decision-maker, when presented with any pair of actions,

knows which of the pair she prefers (Completeness), or knows that sheregards both actions as equally desirable (is ”indifferent between theactions”).

- Rational Choice (Transitivity): We assume further that these preferences areconsistent in the sense that if the decision-maker prefers the action a to theaction b, and the action b to the action c, then she prefers the action a tothe action c.

- Note that we do not rule out the possibility that a person’s preferences arealtruistic in the sense that how much she likes an outcome depends on someother person’s welfare.

- We can ”represent” the preferences by a payoff function, which associates anumber with each action in such a way that actions with higher numbers arepreferred1.

a ≺ b ⇐⇒ u(a) < u(b)1 Utility Representation: Refer to Ariel Rubinstein’s Lecture Notes in Microeconomic Theory, Ch2. (Free on his web page!)




A Digression on Transitivity:

- Aggregation of considerations as a source of intransitivity.

a �1 b �1 c;b �2 c �2 a; c �3 a �3 b ⇒ a � b � c � a

- The use of similarities as an obstacle to transitivity.

An individual may express indifference in a comparison between two elementsthat are too ”close” to be distinguishable.

I The theory of rational choice is that in any given situation the decision-makerchooses the member of the available subset of A that is best according to herpreferences.

I The action chosen by a decision-maker is at least as good, according to herpreferences, as every other available action.


The Extensive Form Representation of a Game





What is a Game?

I From the noncooperative point of view, a game is a multi-person decision situationdefined by its structure, which includes:

- Players: Independent intelligent decision makers

- Rules: Which specify the order of players’ decisions, their feasible decisionsat each point they are called upon to make one, and the information theyhave at such points.

- Actions: All the alternatives available to the players. We call A the set ofactions.

- Outcome: How players’ decisions jointly determine the physical outcome.

- Preferences: players’ preferences over outcomes.



Examples

I Matching Pennies (version A).

Players: There are two players, denoted 1 and 2.

Rules: Each player simultaneously puts a penny down, either heads up or tails up.

Actions: Ai = {H,T}

Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.

I Matching Pennies (version B).

Players: There are two players, denoted 1 and 2.

Rules: Player 1 puts a penny down, either heads up or tails up. Then, Player 2puts a penny down, either heads up or tails up.

Actions: Ai = {H,T}

Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.



Examples

I Matching Pennies (version C).Players: There are two players, denoted 1 and 2.Rules: Player 1 puts a penny down, either heads up or tails up, without lettingplayer 2 know his decision. Player 2 puts a penny down, either heads up or tails up.Actions: Ai = {H,T}Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.

I Matching Pennies (version D).Players: There are two players, denoted 1 and 2.Rules: Players flip a fair coin to decide who begins. The looser puts a penny down,either heads up or tails up. Then, the winner puts a penny down, either heads upor tails up.Actions: Ai = {H,T}Outcomes: If the two pennies match, the looser pays 1 dollar to player 2;otherwise, the winner pays 1 dollar to player 1.




I Some games that are important in economics have simultaneous moves.

I ”Simultaneous” means strategically simultaneous, in the sense that players’decisions are made without knowledge of others’ decisions.

I It need not mean literal synchronicity, although that is sufficient for strategicsimultaneity.

I But many important games have at least some sequential decisions, with somelater decisions made with knowledge of others’ earlier decisions.

I We need a way to describe and analyze both kinds of game.

I One way to describe either kind of game is via the extensive form or game tree,which shows a game’s sequence of decisions, information, outcomes, and payoffs.




I A version of Matching Pennies with sequential decisions, in which Player 1 movesfirst and player 2 observes 1’s decision before 2 chooses his decision.




I We can represent the usual Matching Pennies with simultaneous decisions byintroducing an information set, which includes the decision nodes a player cannotdistinguish and at which he must therefore make the same decision, as in thecircled nodes.




I The order in which simultaneous decision nodes are listed has some flexibility, as inprevious case, where player 2 could have been at the top.

I For sequential decisions the order must respect the timing of information flows.(Information about decisions already made, as opposed to predictions of futuredecisions, has no reverse gear.)

I All decision nodes in an information set must belong to the same player and havethe same set of feasible decisions. (Why?)

I Players are normally assumed necessarily to have perfect recall of their own pastdecisions (and other information). If so, the tree must reflect this.

DefinitionA game is one of perfect information if each information set contains a single decisionnode. Otherwhise, it is a game of imperfect information.




I This is an example of a game with simultaneous decision nodes and players withperfect recall of their own past decisions.




I This is an example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.




I This is another example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.


Strategies and the Normal Form Representation of a Game






I For sequential games it is important to distinguish strategies from decisions oractions.

I A strategy is a complete contingent plan for playing the game, which specifies afeasible decision for each of a player’s information sets in the game.

I Recall that his decision must be the same for each decision node in an informationset.

I A strategy is like a detailed manual of actions, not like a single decision or action.




I It is assumed that conditional on what a player observes, he can predict theprobability distributions of his own and others’ future decisions and theirconsequences.

I If players have this kind of foresight, then their rational sequential decision-makingin ”real time” should yield exactly the same distribution of decisions assimultaneous choice of fully contingent strategies at the start of play.

I The player writes his own manual of actions. Then he will give you (a neutralreferee) the manual and let you play out the game. You will tell him who won.

I Because strategies are complete contingent plans, players must be thought of aschoosing them simultaneously (without observing others’ strategies),independently, and irrevocably at the start of play.




Player 2 strategies:

I Strategy 1 (s1): Play H if player 1 plays H; Play H if player 1 plays T

I Strategy 2 (s2): Play H if player 1 plays H; Play T if player 1 plays T

I Strategy 3 (s3): Play T if player 1 plays H; Play H if player 1 plays T

I Strategy 4 (s4): Play T if player 1 plays H; Play T if player 1 plays T




I A game maps strategy profiles (one for each player) into payoffs (with outcomesimplicit).

I There is a one-to-one correspondence between actions and outcomes. Thereforewe can consider the preferences, and payoffs, to be over strategies.

I Specifying strategies make it possible to describe an extensive-form game’srelationship between strategy profiles and payoffs by its (unique) normal form orpayoff matrix or (usually when strategies are continuously variable) payoff function.




I The mapping from the normal to the extensive form isn’t univalent: the normalform for Matching Pennies version B has possible extensive forms other than theone depicted before:



Randomized Choices

I In game theory it is useful to extend the idea of strategy from the unrandomized(pure) notion we have considered to allow mixed strategies (randomized strategychoices).

I Example: Matching Pennies Version A has no appealing pure strategies, but thereis a convincingly appealing way to play using mixed strategies: randomizing 50-50.(Why?)

I Our definitions apply to mixed as well as pure strategies, given that theuncertainty about outcomes that mixed strategies cause is handled (just as forother kinds of uncertainty) by assigning payoffs to outcomes so that rationalplayers maximize their expected payoffs.

I Mixed strategies will enable us to show that (reasonably well-behaved) gamesalways have rational strategy combinations.


The Rational Choice Paradigm






I The assumption that the player is rational lies at the foundation of what is knownas the rational choice paradigm.

Definition

Rational Choice AssumptionsThe player fully understands the decision problem by knowing:

I The set of all possible actions AI The payoffs associated whit the strategies R.I Exactly how each strategy affects which outcome will materializeI Her rational preferences over payoffs

Rational Player: A player facing a decision problem that will maximizes his payoffover all possible strategies.




I A rational player chooses the action that brings her the highest payoff

I The rational player will have the most fun


Exercises




Exercises

I Exercise 1. (Altruistic preferences) Person 1 cares both about her income andabout person 2’s income. Precisely, the value she attaches to each unit of her ownincome is the same as the value she attaches to any two units of person 2’sincome. How do her preferences order the outcomes (1, 4), (2, 1), and (3, 0),where the first component in each case is person 1’s income and the secondcomponent is person 2’s income? Give a payoff function consistent with thesepreferences. Is that payoff unique?

I Exercise 2. (Alternative representations of preferences) Adecision-maker’spreferences over the set A = a, b, c are represented by the payoff function uforwhich u(a) = 0, u(b) = 1, and u(c) = 4. Are they also represented by thefunction v for which v(a) = -1, v(b) = 0, and v(c) = 2? How about the functionw for which w(a) = w(b) = 0 and w(c) = 8?

I Exercise 3. Deffine x ∼ y ⇐⇒ x � y and y � x . Denote the set of acctions byX . Define I(x) to be the set of all y in X for which y ∼ x . Show that the set (ofsets!) I(x)|x ∈ X is a partition of X . That is:

For all x and y , either I(x) = I(y) or I(x) ∩ I(y) = ∅.For every x ∈ X , there is y ∈ X such that x ∈ I(y).


Exercises

I Exercise 4. In a game where player i has N information sets indexed n = 1, · · · ,Nand Mn possible actions at information set n, how many strategies does player ihave?

I Exercise 5. Depict the normal formm of Matching Pennies Version C.


Exercises

I Exercise 6. Consider the followign two-player (excluding payoffs):

a) What are player 1’s possible strategies? player 2’s?

b) Suppose that we change the game by merging the information set of player 1’ssecond round of moves (so that all the four nodes are now in a single informationset). Argue why the game is no longer one of perfect recall.


Formalizing the Game






I Up to this point we defined game without been formal. Let me introduce someNotation:

- set of players: I = {1, 2, · · · ,N}

- set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai .

- strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set ofpure strategies Si available to him. A strategy is a complete contingent planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si .

Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i ) ∈ (Si , S−i ).

- Payoff function: ui :∏N

i=1 Si → R, denoted by ui (si , s−i )




I Now we can denote game with pure strategies and complete information in normalform by: ΓN = {I, {Si}i , {ui}i}.

I What about the games with mix strategies?

I We have taken it that when a player acts at any information set, hedeterministically picks an action from the set of available actions. But there is nofundamental reason why this has to be case.

DefinitionA mixed strategy for player i is a function σi : Si → [0, 1], which assigns a probabilityσi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying

∑si∈Si

σi (si ) = 1.

I We denote the set of mixed strategies by ∆(Si ).

I Note that a pure strategy can be viewed as a special case of a mixed strategy inwhich the probability distribution over the elements of Si is degenerate.



Example

I Meeting in New York:

- Players: Two players, 1 and 2

- Rules: The two players can not communicate. They are suppose to meet in NYCat noon to have lunch but they have not specify where. Each must decide whereto go (only one choice). Art Museum vs. Brooklyn Bridge for player 1. ArtMuseum vs. Brooklyn Bridge vs. Central Park for player 2.

- Outcomes: If they meet each other, they enjoy other’s company. Otherwise, theyeat alone.

- Payoffs: They attach a monetary value of 100 USD to other’s company and 0USD to eat alone.

player 2A B C

player 1A 100,100 0,0 0,0B 0,0 100,100 0,0



Example

I Meeting in New York:

- set of players: I = {1, 2}

- set of actions: A1 = {A,B}, and A2 = {A,B,C}

- strategies for each player: S1 = A1, and S2 = A2 (Why?)

- Payoff function: ui :∏2

i=1 Si → R, denoted by ui (si , s−i )

u(si , s−i ) ={

1000

if si = s−i

if si 6= s−i

I Player 2

- pure strategies: S2 = {A,B,C}. Player 2 has 3 pure strategies.

- mixed strategies:∆(S2) = {(σ2

1 , σ22 , σ

23) ∈ R3|σ2

m ≥ 0∀m = 1, 2, 3 and∑3

m=1 σ2m = 1}


Dominant and Dominated Strategies





Dominant and Dominated StrategiesI Now we turn to the central question of game theory: What should be expected to

observe in a game played by rational agents who are fully knowledgeable about thestructure of the game and each others’ rationality?

I To keep matters simple we initially ignore the possibility that players mightrandomize in their strategy choices.

I The prisoner’s dilemma:* There are two thieves who are suspects in an armed robbery. Each thieve is in solitary

confinement with no means of speaking to or exchanging messages with the other.* The prosecutors do not have enough evidence to convict the pair on the principal charge.

To nail the suspects for the robbery, the police need testimony from at least one of thesuspects.

* Simultaneously, the prosecutors offer each prisoner a deal that reduces the sentence hewill get if he confesses on his partner in crime.

* The alternative is for the suspect to say nothing to the investigators.* Here is the offer:

- If both choose to remain silent, then both get 2 years in prison because the evidencecan support only the charge of petty theft.

- If player 1 remains silent while player 2 confesses, then player 1 gets 10 years inprison while player 2 gets only 1 year for being cooperative and vice versa.

- If both confess then both get 9 years in prison.




I Let me put prisoner’s dilemma as a game of trust:

player 2silent confess

player 1silent -2,-2 -10,-1

confess -1,-10 -9,-9

I Observe that regardless of what her opponent does, player i is strictly better offplaying confess rather than silent. This is precisely what is meant by a strictlydominant strategy.

- Player 2 plays silent. Player 1 knows that −1 > −2, better to confess.

- Player 2 plays confess. Player 1 knows that −9 > −10, better to confess.

- Regardless of the other’s strategies, it is always better to confess.

- Note that both would be better off if they both play silent.

Lesson self-interested behavior in games may not lead to socially optimal outcomes.




DefinitionA strategy si ∈ Si is a strictly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) > ui(s̃i , s−i).

I A strictly dominant strategy for i uniquely maximizes her payoff for any strategyprofile of all other players.

I If such a strategy exists, it is highly reasonable to expect a player to play it. Thisis a consequence of the Rational Choice Paradigm.




I What about if a strictly dominant strategy doesn’t exist?

player 2a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

I You can easily convince yourself that there are no strictly dominant strategies herefor either player.

I Notice that regardless of whether Player 1 plays A or B, Player 2 does strictlybetter by playing b rather than a.

I That is, a is ”strictly dominated” by b.




DefinitionA strategy si ∈ Si is strictly dominated for player i if there exists astrategy s̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) > ui(si , s−i). In thiscase, we say that s̃i strictly dominates si .

I In words, s̃i strictly dominates si if it yields a strictly higher payoff regardless ofwhat (pure) strategy rivals use.

I Note that the definition would also permits us to use mixed strategies.

I Using this terminology, we can restate the definition of strictly dominant: Astrategy si is strictly dominant if it strictly dominates all other strategies.

I It is reasonable that a player will not play a strictly dominated strategy, aconsequence of rationality, again.


Iterated Delation of Strictly Dominated Strategies





Iterated Delation of Strictly Dominated Strategiesplayer 2

a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

I We argued that a is strictly dominated (by b) for Player 2; hence rationality ofPlayer 2 dictates she won’t play it.

I We can push the logic further: if Player 1 knows that Player 2 is rational, heshould realize that Player 2 will not play strategy a.

I Notice that we are now moving from the rationality of each player to the mutualknowledge of each player’s rationality.

I Once Player 1 realizes that 2 will not play a and ”deletes” this strategy from thestrategy space, then strategy A becomes strictly dominated by strategy B forPlayer 2.

I If we iterate the knowledge of rationality once again, then Player 2 realizes that 1will not play A, and hence ”deletes” A.

I Player 2 should play c. We have arrived at a ”solution”.




DefinitionA game is strict-dominance solvable if iterated deletion of strictlydominated strategies results in a unique strategy profile.

I Since in principle we might have to iterate numerous times in order to solve astrict-dominance solvable game, the process can effectively can only be justified bycommon knowledge of rationality.

I As with strictly dominant strategies, it is also true that most games are notstrict-dominance solvable.

I You might worry whether the order in which we delete strategies iterativelymatters. Insofar as we are working with strictly dominated strategies so far, it doesnot.


Iterated Delation of Dominated Strategies






DefinitionA strategy si ∈ Si is a weakly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) ≥ ui(s̃i , s−i), and for at least onechoice of s−i the inequality is strict.

DefinitionA strategy si ∈ Si is weakly dominated for player i if there exists a strategys̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) ≥ ui(si , s−i), and for at leastone choice of s−i the inequality is strict. In this case, we say that s̃i weaklydominates si .

DefinitionA game is weakly-dominance solvable if iterated deletion of weaklydominated strategies results in a unique strategy profile.




I Using this terminology, we can restate the definition of weakly dominant: Astrategy si is weakly dominant if it weakly dominates all other strategies.

I You might worry whether the order in which we delete strategies iterativelymatters. Delation of dominated strategies could leave to different outcomes.

P2L R

U 5,1 4,0P1 M 6,0 3,1

D 6,4 4,4

P2L R

P1U 5,1 4,0D 6,4 4,4

P2L R

P1M 6,0 3,1D 6,4 4,4


Exercises




Exercises

ExercisesI Exercise 7. Prove that a player can have at most one strictly dominant strategy.I Exercise 8. Apply the iterated elimination of strictly dominated strategies to the

following normal form games. Note that in some cases there may remain morethat one strategy for each player. Say exactly in what order you eliminated rowsand columns.

I Exercise 9. Apply the iterated elimination of dominated strategies to the followingnormal form games. Note that in some cases there may remain more that onestrategy for each player. Say exactly in what order you eliminated rows andcolumns.


Exercises

Exercises

I Exercise 8 and 9 (cont.).


Exercises

Exercises



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