An introduction to game theoryComing soonPreviouslyTodayWho is this?Three elements in every gameGame theory: Payoff matrixHow do we interpret this box?Slide 9Back to a Core Principle: EquilibriumHow do we find Nash equilibrium (NE)?Steps 1 and 2Step 3:Step 4Step 5Step 6Double check our NESlide 18Dominant strategyNew exampleSlide 21Two NE possibleSlide 23Slide 24Sequential decisionsDecision tree in a sequential game: Person 1 chooses firstSlide 27Slide 28Slide 29More on game theorySummaryNext time…An introduction to game theoryToday: The fundamentals of game theory, including Nash equilibriumComing soonTest 2Practice problems are posted on class websiteIf you would like help preparing for Test 2, please ask in office hours and/or during sectionMost questions are difficult to answer by e-mailI will also use end of lectures for questions, as time permitsPreviouslyMarkets with perfect competitionEfficiency with the “invisible hand”Monopolistic marketsInefficient outcomes without regulationRegulation does not guarantee efficiencyTodayIntroduction to game theoryWe can look at market situations with two players (typically firms)Although we will look at situations where each player can make only one of two decisions, theory easily extends to three or more decisionsWho is this?Find out who this early researcher in game theory is in classThree elements in every gamePlayersTwo or more for most games that are interestingStrategies available to each playerPayoffsBased on your decision(s) and the decision(s) of other(s)Game theory: Payoff matrixA payoff matrix shows the payout to each player, given the decision of each playerAction C Action DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2How do we interpret this box?The first number in each box determines the payout for Person 1The second number determines the payout for Person 2Action CAction DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2How do we interpret this box?ExampleIf Person 1 chooses Action A and Person 2 chooses Action D, then Person 1 receives a payout of 8 and Person 2 receives a payout of 3 Action CAction DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Back to a Core Principle: EquilibriumThe type of equilibrium we are looking for here is called Nash equilibriumNash equilibrium: “Any combination of strategies in which each player’s strategy is his or her best choice, given the other players’ choices” (F/B p. 322)Exactly one person deviating from a NE strategy would result in the same payout or lower payout for that personHow do we find Nash equilibrium (NE)?Step 1: Pretend you are one of the playersStep 2: Assume that your “opponent” picks a particular actionStep 3: Determine your best strategy (strategies), given your opponent’s actionUnderline any best choice in the payoff matrixStep 4: Repeat Steps 2 & 3 for any other opponent strategiesStep 5: Repeat Steps 1 through 4 for the other playerStep 6: Any entry with all numbers underlined is NESteps 1 and 2Assume that you are Person 1Given that Person 2 chooses Action C, what is Person 1’s best choice?Action CAction DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Step 3:Underline best payout, given the choice of the other player Choose Action B, since 12 > 10  underline 12Action CAction DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Step 4Now assume that Person 2 chooses Action DHere, 10 > 8  Choose and underline 10Action C ActionDAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Step 5Now, assume you are Person 2If Person 1 chooses A3 > 2  underline 3If Person 1 chooses B4 > 1  underline 4Action C Action DActionA10, 2 8, 3ActionB12, 4 10, 1Person 1Person 2Step 6Which box(es) have underlines under both numbers?Person 1 chooses B and Person 2 chooses CThis is the only NEAction C Action DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Double check our NEWhat if Person 1 deviates from NE?Could choose A and get 10Person 1’s payout is lower by deviating Action C Action DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Double check our NEWhat if Person 2 deviates from NE?Could choose D and get 1Person 2’s payout is lower by deviating Action C Action DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2Dominant strategyA strategy is dominant if that choice is definitely made no matter what the other person choosesExample: Person 1 has a dominant strategy of choosing BAction C Action DAction A10, 2 8, 3Action B12, 4 10, 1Person 1Person 2New exampleSuppose in this example that two people are simultaneously going to decide on this gameYes NoYes 20, 20 5, 10No 10, 5 10, 10Person 1Person 2New exampleWe will go through the same steps to determine NEYes NoYes 20, 20 5, 10No 10, 5 10, 10Person 1Person 2Two NE possible(Yes, Yes) and (No, No) are both NEAlthough (Yes, Yes) is the more efficient outcome, we have no way to predict which outcome will actually occurYes NoYes 20, 20 5, 10No 10, 5 10, 10Person 1Person 2Two NE possibleWhen there are multiple NE that are possible, economic theory tells us little about which outcome occurs with certaintyTwo NE possibleAdditional information or actions may help to determine outcomeIf people could act sequentially instead of simultaneously, we could see that 20, 20 would occur in equilibriumSequential decisionsSuppose that decisions can be made sequentiallyWe can work backwards to determine how people will behaveWe will examine the last decision first and then work toward the first decisionTo do this, we will use a decision treeDecision tree in a sequential game: Person 1 chooses firstABCPerson 1 chooses yesPerson 1 chooses noPerson 2 chooses yesPerson 2 chooses yesPerson 2 chooses noPerson 2 chooses no20, 205, 1010, 510, 10Decision tree in a sequential game: Person 1 chooses firstGiven point B, Person 2 will choose yes (20 > 10)Given point C, Person 2 will choose no (10 > 5)ABCPerson 1 chooses yesPerson 1 chooses noPerson 2 chooses yesPerson 2 chooses yesPerson 2 chooses noPerson 2 chooses no20, 205, 1010, 510, 10Decision tree in a sequential game: Person 1 chooses firstIf Person 1 is rational, she will ignore potential choices that Person 2 will not makeExample: Person 2 will not choose yes after Person 1 chooses noABCPerson 1 chooses