Term 2 Week 6: Introduction to Game Theory and Nash Equilibrium Flashcards
What is the difference between a strategic and non-strategic action (2)
-A strategic action is one where other peoples actions impact you (an auction)
-A non strategic action is one where other peoples actions don’t impact you (what to buy at a supermarket)
What must we define for game theory (3)
-Parties with decisions to make (Players = N)
-Different choices to make (Actions, Ai)
-Outcomes for each combination of choices (Payoffs, π:A1 x A2)
What does a payoff matrix look like (3)
-You have a table with Player 1’s actions on the rows, Player 2’s actions on the columns
-Within the table, each square represents the payoffs for the 2 players
-The number on the left represents the row players payoff, the player on the right represents the column players payoff
What are the 3 problems for players in game theory, and the assumption made about payoffs (3,1)
Players must:
-Understand the game
-Form expectations on what the other placer will do
-Find the best response to what you anticipate the other players doing
-It is assumed payoffs must encode everything (long run, risk appetite) the players care about, and thus players are modelled to maximise their payoff
What is a solution concept (1,2)
-A solution concept narrows down the set of outcomes to reasonable ones
-A solution concept is a way to make better predictions about how people will play (positive)
-A solution concept is a way to give better advice about how to play the game (normative)
what is a best response (4)
-An action a(i) is a best response if it yields the highest payoff, given the opponents action profile a(-i)
-Formally, a(I) is a best response to profile a(-i) if: a(I) = arg max π(a(I), a(-i))
-To form a best response, players must predict what other players do, and decide what their best action is in response
-This is the basis behind Nash Equilibrium
What is a dominant strategy + example (4,2)
-If there is an action which is always the best response, this is the dominant strategy
-This is strongly dominant if its always the best (π(a(I), a(j)) > π(a’(i), a(j)) for all a(j))
-This is weakly dominant if it is the best response for at least one profile, and at least the same for the others (π(a(I), a(j)) ≥ π(a’(i), a(j)) for all a(j), (π(a(I), a(j)) > π(a’(i), a(j)) for some a(j))
-With dominant strategies, we can also say what not to pick, and a’ is dominated if it always gives a lower payoff for any a(j)
-Imagine the prisoners dilemma (Confess vs Quiet), where (C,C) = 1,1, (C,Q) = 10, 0, (Q,C) = 0, 10, (Q,Q) = 5,5
-The dominant strategy is to confess, no matter what (look at if row player C or Q then col players best choice)
What is the Iterated Elimination of Dominated Strategies (2,2,4)
-With multiple options, you can have a dominated strategy w/o a dominant one
-You could then assume players should never expect others to play dominated strategies, or even never ones which are never best responses
-Repeatedly removing all dominated actions to see which remain is called IEDS
-After this process we are left with rationalizable strategies
-Imagine a 3x3 payoff matrix where 2 students can put no effort, some effort or high effort into a test
-(N,N) = 0,0, (N,S) = -100, 300, (N,H) = -100, 150, (S,N) = 300, -100, (S,S) = 100,100, (S,H) = -50, 150, (H,N) = 150, -100, (H,S) = 150, -50, (H,H) = 50,50
-In this case you can eliminate No Effort as a strategy, as it is dominated
-Even if you changed (S,N) to 300, 0 so that it was no longer dominant, it is still not rationalizable as it is never the best outcome
Do rationalizable outcomes have to be rational (2)
-Rationalizable outcomes include those where players miscoordinate (both go or stop at a stop light)
-Players select the best response to the expected action, but not the actual action, and they’d want to go back and change their strategies
What is Nash Equilibrium (2)
-An action profile is a Nash Equilibrium if all players are playing best responses
-Formally, a profile (a(I), a(-i) is a Nash equilibrium if a(I) ε arg max π(a(i), a(-i)) for all i ε N
What are the 3 different interpretations of Nash Equilibrium (3,1)
-One way to think of Nash Equilibrium is a self-enforcing outcome, where no individual has an incentive to change their actions
-Another way of interpreting Nash Equilibrium is the resting point of a dynamic process of adjustment, where we update our best response over time in trial and error fashion
-A third way of interpreting Nash equilibrium is as an outcome arrived at by rational introspection, with other outcomes being hard to justify if all players are rational
-Interpretations 2 and 3 don’t apply to all games, only ones with dominated strategies (imagine rock paper scissors)
Why should peoples beliefs be correct, and why is this important for Nash Equilibrium (3,1)
Peoples beliefs should be correct since:
-Expectations get updated over time
-Players will have reputations to take certain actions
-Tacit coordination (focal points)
-It is important for peoples beliefs to be important, as players best responding to these beliefs thus means Nash Equilibria are stable
How do we solve games with multiple pure strategy nash equilibria (1,3)
-Games with multiple NE present a problem, as we don’t know which one will actually happen, and people could get stuck in bad equilibria
We can solve this via:
-Mediation (have a third party control the game)
-Conventions (unspoken/spoken rules)
-Focal points (out of many NE which will happen naturally)
What is Thomas Schnelling’s idea of focal points (2,1)
-Thomas Schnelling’s idea of focal points suggest that some NE are more natural than others
-They are culturally more salient/focal than others, and crucially this must be common knowledge
-This is different from conventions, that are rules/norms that evolve over time
What are pure strategies (2)
-Nash equilibria are stable points of the game, but not all games have Nash equilibrium, at least not in pure strategies
-Pure strategies means the players only pick one strategy, and there’s no outcome where all the players wouldn’t want to pick their strategy
What is the concept of a mixed strategy (2,1)
-The concept of a mixed strategy is the idea where players randomise between pure strategies
-In this game, the players want to be unpredictable, so randomising makes sense
-Although it can then be asked if threr are a nash equilibrium frequency of strategies
How do we work out the best response for a driver/police if q = probability of monitoring, p = probabilty of speeding, and the payoffs for the driver/police are (-50,70) (Sp, Mon), (20,-10) (Sp, DM), (0,5) for (DS, M) and (0,15) for (DS, DM) (3,3)
-The payoff for speeding will be: πD (speed) = q(-50) + (1-q)(20) = 20-70q
-The payoff for not monitoring will be: πD(don’t) = 0
-Equalling these to eachother, the driver should speed if q < 2/7
-The payoff for monitoring will be: πP (monitor) = p(70) + (1-)(5) = 65p + 5
-The payoff for not monitoring will be: πP(don’t) = p(-10) + (1-p)(15) = 15-25p
-Equalling these to eachother, the driver should speed if p>1/9
How do we draw best response diagrams for the driver, police and both if q = probability of monitoring, p = probabilty of speeding, and the payoffs for the driver/police are (-50,70) (Sp, Mon), (20,-10) (Sp, DM), (0,5) for (DS, M) and (0,15) for (DS, DM) (3,3,3)
-For the driver, have q on the x axis and πD on the y axis
-πD (Don’t) will be a horizontal line on the x axis, πD (speed) will be a linear downward sloping line with a y intercept 20 and y value -50 when q = 1
-The intersect of the 2 lines is 2/7, to the left shows speeding giving a higher payoff, to the right shows not giving a higher payoff
-For the police, have p on the x axis and πP on the y axis
-πD (monitor) will be an upward sloping linear line with coordinates (0, 5) and (1, 70), and πD (don’t) will be a downward sloping linear line with coordinates (0, 15) and (1, -10)
-The intersect of the 2 lines is 1/9, to the left the police shouldn’t monitor, to the right they should
-Now draw a diagram with p on the x axis and q on the y axis
-BRp should be 0 for q until p = 1/9, then jump up to q = 1 onwards
-BRD should be at p = 0 from q = 1 to q = 2/7, and p = 1 from q = 2/7 to q = 0
What happens if a finite game has no pure strategy nash equilibrium (1)
-There always is a mixed strategy nash equilibrium
What are 2 interpretations of mixed strategies offered by Nash ()
-Players randomise over strategies in 2 player games
-Mass-action interpretation in N-person games
What is the payoff matrix for the stag game (1, 4, 2)
-Both players have the choice of either hunting a stag or hare
-P(S, S) = 5,5
-P(S, H) = 0,3
-P(H,S) = 3,0
-P(H,H) = 3,3
-In this game it benefits you to play the same strategy as the other person
-This is a game of trust and cooperation, with 2 Nash Equilibrium
What situations can the stag hunt game applied (3)
-There is an outcome which is Pareto superior to all others (known as the payoff dominant outcome)
-The payoff dominant outcome is stable (NE) but requires coordination
-There is uncertainty due to a trade-off between mutual benefit and individual risk
Why is the stag hunt game different to the prisoners dilemma (2)
-Cooperation doesn’t give the highest individual payoff in the PD
-The cooperative outcome is not stable (NE) in the PD
How can we adapt a 2 player game to a n player game, with hold (1-p) and withdraw (p), as options, where (H,H) = (12,12), (H, W) = (0, 10), (W, H) = (10,0), (W,W) = (5,5) (3,2)
-First draw the diagram with p on the x axis, and both hold and withdraw
-Say we start at p = 0.7, so the payoff from withdrawing > payoff from holding
-If we allow players to adapt, they will start withdrawing and p will rise until we get to p = 1, where no one has an incentive to change
-If we start at p = 2/7, this is a stable equilibrium (mixed strategy equilibrium)
-This is less stable, as if a few people changed action, we would start to converge to a different stable equilibrium (p = 0 or p = 1)
What are other applications of the stag hunt game (4)
-Industrial action/protests
-Firms hiring in recessions
-Peer effects
-Panic buying
How can we model panic buying in a market with a stag hunt game (4)
-Assume don’t panic is the stag, panic is the hare
-(DP, DP) = (a,a), (DP, P) = (c,b), (P, DP) = (b,c), (P,P) = (d,d)
-Assume a>b, d>c, a>d
-There are 2 nash equilibria in pure strategies (a,a), and one is pareto efficient (a,a)
-Imagine we play this game against N > 2 players, where p is the proportion playing don’t panic
-The expected payoff from don’t panic is: π(D) = p(a) + (1-p)c
-The expected payoff from panic is π(D) - pb + (1-p)d
-When drawing our expected payoff diagram, we can see if maximally uncertain (p = 1/2) then hare is the best response strategy (risk dominant strategy)
-p̅ is the tipping point, where the 2 curves meet, describing which equilibria is more likely to rise
-c is the risk of playing stag, lowering c makes stag more risky, meaning p̅ rises (have to be even more convinced of others playing stag to play stag)
-lowering c can be seen by keeping the same point at p = 1, but lowering p = 0, pivoting the stag graph around the p = 1 point
How can stock outs arise endogeneously (3)
-Stock outs can arise endogenously (as a NE) when there is no change in the actual supply of the product
-It can be a best response whenever your belief about p is sufficiently low
-Panic buying can thus become a self-fufilling prophecy
When is panic buying more likely + the governments role (1,2)
-Panic buying is more likely is p̅ is high
p̅ is high when:
-c is low (payoff from not panic buying when others do) (fuel, as you need it)
-When b is high (the cost of panic buying is low) (tinned products, which could be stored)
-The governments role is to maintain a high belief p (p = belief of playing stag) and minimise uncertainty
