ECON-345: Strict Dominance (Aug 29, 2024) PDF

ECON-345: August 29, 2024 Making predictions by ruling out dominated strategies Agenda 1. Review: key points from Chapter 2. 2. Predicting outcomes in strategic situations: a heuristic overview 3. Definition: “strictly dominate” 4. Example: prisoner’s dilemma 5. Another perspective on Coke...

ECON-345: August 29, 2024 Making predictions by ruling out dominated strategies Agenda 1. Review: key points from Chapter 2. 2. Predicting outcomes in strategic situations: a heuristic overview 3. Definition: “strictly dominate” 4. Example: prisoner’s dilemma 5. Another perspective on Coke and Pepsi 6. Add one more assumption: I know that you’re rational. (Big pig, little pig.) Extras Weak dominance Randomizing between two strategies can strictly dominate a third strategy Review of Chapter 2 Key elements of a game: Players Strategy sets These elements are common Payoff functions knowledge to the players. (Timing of actions & info sets) (More on this later…) Two ways to represent a game: Strategic form. (List players, strategy sets, payoff functions.) Often display in a matrix if two players and only a few strategies. Game tree. (aka “extensive form”) More useful if there is a sequence of events that needs to be conveyed.  Example: a class walk-out game Student Student Stay22 Go2 Sta 1 Go1 3.3 , y12 Student Student 2 Student Stay1 4,0 3.3 Sta Sta Go2 Go2 3.7 , y2 y2 Go1 0,4 1 3.7 3.3 , 4 , 00 , 4 3.7 , 3.3 3.7 The diagrams on the left and the right do not represent the same strategic situation. Why?  Vocab: what is Student 2’s strategy set in each game below? Student Student Stay22 Go2 Sta 1 Go1 3.3 , y12 Student Student 2 Student Stay1 4,0 3.3 Sta Sta Go2 Go2 3.7 , y2 y2 Go1 0,4 1 3.7 3.3 , 4 , 00 , 4 3.7 , 3.3 3.7 Strategy: Complete plan for how to act at each point in the game where I might have to make a decision. Game on the left: Game on the right:  We can revise the game tree so that it does represent exactly the same strategic situation as the matrix on the left. (How?) Student Student Stay22 Go2 Sta 1 Go1 3.3 , y12 Student Student 2 Student Stay1 4,0 3.3 Sta Sta Go2 Go2 3.7 , y2 y2 Go1 0,4 1 3.7 3.3 , 4 , 00 , 4 3.7 , 3.3 3.7 Key concept: a player’s information set(s). (More on this later.) The rest of this class: strictly dominated strategies What does it mean to say one strategy is strictly dominated by another? How do we identify strictly dominated strategies? And why do we rule them out when making predictions about how the game will be played? Predicting outcomes in strategic situations (Stronger assumptions  sharper predictions.) Implication about the strategies I might We will always assume: use All players are rational.  We will then impose stronger and stronger assumptions about beliefs. What do players believe about how their opponents will act? Believe other players are rational too.  Believe others are rational and know that I am rational.  Correct beliefs about how others will act.  Definition of “strictly dominate” [In the case where there are two players and and are strategies for Player 1] strictly dominates for Player 1 if holds for every possible strategy by Player 2. Claim: Stay1 strictly dominates Go1 for Player 1 in the e.g. from earlier. If Player 2 plays Stay2 If Player 2 plays Go2 : : Student 2 Student 2 Confirm this by comparing P1’s Stay, Go payoffs for every possible strategy Stay2 Go2 Stay2 Go2 of P2. 3.3 , 3.3 , Student Student Stay1 4,0 Stay1 4,0 3.3 3.3 3.7 , 3.7 , Go1 0 , 4 Go 0 , 4 1 1 3.7 1 3.7  We can also show that Stay2 strictly dominates Go2 for Player 2. Student Stay22 Go2 3.3 , If Player 1 Stays: 3.3 Stay1 Student 4,0 3.3 > 0 3.7 , Go1 0,4 1 3.7 Student Stay22 Go2 3.3 , Stay1 Student 4,0 3.3 3.7 , Go1 0,4 If Player 1 Goes: 4 > 1 3.7 3.7 In some games, ruling out strictly dominated strategies suffices to get a sharp prediction for the outcome. Same example: Student Stay22 Go2 3.3 , Student Stay1 4,0 3.3 3.7 , Go1 0,4 1 3.7 Def. A player has a dominant strategy if: But usually, ruling out strictly dominated strategies is just a way to narrow down our set of predictions. We will need to add more assumptions to get a sharp prediction. A harder example from the online module Any strictly dominated strategies for either player?  Real Estate Bubble Game (strategies are “exit the market” times) Backstory? Investor 2 Time 0 Time 1 Time 2 Time 3 Time 0 0,0 0 , -1 0 , -2 0 , -3 Investor Time 1 -1 , 0 -1 , -1 10 , -2 10 , -3 Time 2 -2 , 0 -2 , 10 -2 , -2 20 , -3 1 Time 3 -3 , 0 -3 , 10 -3 , 20 -3 , -3 Based on the backstory, which strategies seem like a bad idea? (That’s not proof that they are strictly dominated, but it gives us an idea of where to look.)  Find any strictly dominated strategies for Investor 1 (Investor 2’s payoffs are suppressed to avoid clutter.) Investor 2 Time 0 Time 1 Time 2 Time 3 Time 0 0 0 0 0 Investor Time 1 -1 -1 10 10 Time 2 -2 -2 -2 20 1 Time 3 -3 -3 -3 -3 This is a case where rationality alone (players don’t use SD strategies) narrows the set of possible outcomes only slightly.  How to show one strategy strictly dominates another when strategy sets are continuous: The Cola Wars example (from Canvas) Inverse demand curve: Payoffs Coke: Pepsi: Strategy sets: You showed (graphically) that strictly dominates for Coke. Now we will show algebraically that strictly dominates when is any Coke quantity strictly less than 30.  Start: Get intuition by experimenting with Coke’s payoff function Payoffs Coke: Pepsi: Strategy sets:  Now set up the SD definition algebraically. Payoffs Coke: Pepsi: Strategy sets: Additional assumption: I know that you’re rational Version with facemasks Weak dominance Extras?