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Markets and Strategy Summary
Summary of Markets and Strategy at Radboud University year 3
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Markets and Strategy – Summary of summaryLecture 1; games are everywhere
Non-cooperative game theory: players can’t work together and can’t communicate
- Each player has the properties that they;
o Act indecently (have their own goals)
o Act with self-interest (try to attain their goals)
o Act with an anticipated reaction (they account for other players’ actions)
Games are a collective name to describe models of strategic interaction
1) Map out a real world scenario
2) Determine the appropriate assumptions from you observation
3) Create an analytical model (a game) answering the questions (what are the building blocks?)
- Who are the players?
- What are the actions that players can choose?
- What is the information available to players?
- What are the payoffs?
! interaction flow can be simultaneously (tegelijk) or sequential (na elkaar)
! information flow can be complete or incomplete, perfect or imperfect.
! strategies can be finite or infinite
Strategic interactions have outcomes that are interdependent; what is realised for a single player depends on everybody.
Solving games → looking for solutions (but what is the solution (individual best outcome/social best outcome/lock-in?)
- An important assumption in game theory (and economics) is that players are rational;
o Players maximize what they think is important (and they expect others to be rational too)
- The behaviour of players is captured in strategies; a fully contingent plan for playing the game (considering what you
think others will do).
Normal form games
- The players all move simultaneously (tegelijk)
- No player knows what move the other player is actually making, so payoff functions 𝑢𝑖 : 𝑆1 × ⋯ × 𝑆𝑛 → ℝ
- Every player can make choices from their strategy sets Si; collection of all pure strategies for player i.
- A strategy profile si is an ordered list of all strategies made by players in the game.
o We write s-i ∈ S-i for all strategy profiles of all other players, except i.
- Finally, we need payoff functions (or utility functions) 𝑢𝑖 : 𝑆1 × ⋯ × 𝑆𝑛 → ℝ
o you can rank outcomes for players. If ui (s) > ui (ŝ), then player i prefers the outcome s over ŝ (s ≻ ŝ).
With mixed strategies, player i chooses a probability distribution of σi over all elements of Si
- By ΔSi , we mean the set of all possible mixed strategies for player i.
- These mixed strategies can be used to describe
o How a player might let a randomisation process decide for them (like a coin flip)
o How we think players choose, if we know the entire population distribution
o How a player believes another player might choose if they are inherently uncertain
! If interactions are simultaneous and there is no information asymmetry, then we use normal form games.
If there is a set of choices S, then a utility function on this set represents a preference relation ≻.
- The preference relation gives you an idea how a player ranks the choices, relatively.
- Therefore, utility functions are ordinal. The exact choice of a utility function does not matter, the only thing that
matters is the relative numbers it assigns to possible choices.
- Expected utility EU(𝜇) = ∑ 𝑢(𝑠) ⋅ Pr (𝑠)
o So it is the weighted average utility that you get from a specific gamble
Utility functions;
- The person with a linear utility function is, by definition, risk neutral
- The person with a concave utility function is, by definition, risk averse
o They will need to be compensated extra for the risk that they take
- The person with a convex utility function is, by definition, risk loving.
o they would actually prefer a bet with more risk
Battle of the sexes game: two people need to coordinate into going into a single direction, but their preferences differ
Left Right
Left 4,2 0,0
Right 0,0 2,4
o Naturally, u1 (Left, Left) > u1 (Right, Left) → Player one prefers (Left, Left) over (Right, Left).
o Similarly, u1 (Right, Right) > u1 (Left, Right)→ Player one prefers (Right, Right) over (Left, Right)
, - It’s clear what player 1 would do if actions of player 2 are known, but what if there is uncertainty about it?
o Say that player 1 believes that player 2 will randomise. Left or right gets equal chance σ2 =( ½, ½ )
1 1 1 1
▪ u1 (Left, σ2 ) = ⋅ u1 (Left, Left) + ⋅ u1 (Left, Right) = ⋅ 4 + ⋅ 0 = 2
2 2 2 2
1 1 1 1
▪ u1 (Right, σ2 ) = ⋅ u1 (Right, Left) + ⋅ u1 (Right, Right) = ⋅ 0 + ⋅ 2 = 1.
2 2 2 2
o Conclusion: player 1 will prefer to go for Left under the belief σ2 .
The prisoner’s dilemma; a game illustrating conflict between individual and group rationality.
- Two suspects are asked seperately to betray the other one and provide evidence about a
crime they both committed. If only one confesses, the confessor goes free and the other
will get maximally sentenced. If both confess, they provide evidence on each other, and both get badly punished. If they
both remain loyal, they get punished minimally.
The zero-sum game; represents strictly competitive scenarios
- There cannot be a winner, without there being an equal loser
The coordination game; both players want to align their strategies with the other player, but the lack
of good communication will lead to miscoordination in practice
- Examples: driver’s having to give right of way at a road narrowing, passing each other in a
narrow hallway
The game of chicken; both players want to be seen as the ultimate alpha badass. Sadly both trying
to attain that position leads to mutual destruction in the process
- The game of chicken: who will steer away from the conflict first?
The game of dominance; there are two players, one is weaker than the other. The weak player
should remain passive. The dominant player gets a high payoff by not pushing if the weaker player
does, but he will get a higher payoff by pushing if the weaker player does not.
Definitions from Sowiso;
- In a pure coordination game, there are 2 NE, one of which is clearly preferred by both players.
- In the battle of the sexes, there are 2 NE, one of which is clearly preferred by one player and the other NE is preferred
by the other player.
- In a prisoner’s dilemma, there is one NE, which is for both players clearly less beneficial then a non-nash equilibrium
strategy.
- In a zero sum game, the profit of one player is the loss of the other player.
Lecture 2; solutions to simultaneous move games
Two-thirds game; when a player’s payoff depends on predicting the average behaviour of the group.
- All participants simultaneously choose an integer number between 0 and 100. Goal is to pick the number closest to
two-thirds of the groups average. The information structure is complete, but with imperfect information.
- Social dynamics;
o Balancing between Peacocking (standing out) and masking (covering up)
▪ You want to stand out enough to be interesting, but not too much to no longer fit it.
o Your behaviour depends on how you estimate the ‘average’ group behaviour.
- The stock market works like this nowadays;
o The group displays enthousiasm about a company? Then go long;
▪ Purchasing stocks now anticipating that the stocks price will go up and selling them in the future.
o The group displays distrust about a company? Then go short;
▪ Borrow a stock (with a margin requirement as colletaral), sell that borrowed stock at the high price of
the moment, buy it back later at a low price, return it to its rightful owner.
• If the market price goes up after you sell the stock, you end up in a margin call where your
colletaral doesn’t cover your dept.
▪ This may be market breaking in practice; Hedge fund massively shorts a company, then the market
expects that company to drop in value, massive sell-off of the stock by the market, hedge fund has
created a self-fulfilling prophecy.
Best response strategy si* of player i is when ui(si*, s-i) ≥ ui(si, s-i), for all strategies profiles si ∈ ΔSi .
- Best choice correspondence BRi ; maps all best responses of player i for which he is indifferent (~) in choosing (so he
can randomise), taking into account all (beliefs about) strategy choices of players besides i.
o It requires an input (belief about actions of other players) and gives you an ouput (set of all possible responses)
o A best response set is convex; if it contains A and B, then it contains pA + (1- p)B as well.
- Players in games are rational, which means that they will always play their best response, given info about other players
, Example two-thirds game and best response;
If you expect everybody to play 100, then 2/3 is 67, so your best respone is playing 67. If there are people who go below 100, then the average X 2/3 is always
lower than 67, so playing 100 is a dominated strategy since it will always loose out.
- A strategy si is strictly dominated if there is a strategy σi ∈ ΔSi such that ui(σi, s-i) ≥ ui(si, s-i), for all strategies profiles s-i ∈ ΔS-i .
o A strategy is strictly dominated if you can always get a strictly better payoff with another strategy.
o It is a strategy that is always inferior to at least one other strategies that a player can choose.
o A dominated strategy can be removed from the game.
- A strategy is strictly dominant if for all their other strategies σi ∈ ΔSi; ui(si, s-i) > ui(σi, s-i) for all strategies profiles s-i ∈ ΔS-i .
o A dominant strategy is always better than all other possible other possible strategy choices.
Isomorphic games; two games that have the same type of dynamic structure.
A game is common knowledge if it holds that;
- every player knows and understands the game structure;
- every player knows that every other player knows and understands;
- every player knows that the other players know that all players know that they know;
- … and so forth
! if a game is common knowledge, there can be the possibility of iteratively eliminating dominated strategies, so removing a
dominated strategy from a game, then look for dominated strategies in the reduced game, remove those and keep going.
Cognitive hierarchy theory; our rationality depends on the expected depth of strategic thought. How many steps do people
perform in their minds?
- ‘level-zero’ player will choose actions without regard to the actions of other players (=zero-order beliefs)
- ‘level one’ player will choose a best response considering the first-order belief that the other players will act non-
strategically.
- ‘level two’ player acts on the belief that the other players are level one
- Etc. → best repsonse to the previous round average
A game is solved if all players taking the best response to their beliefs. This results in a Nash Equilibrium; a set of actions of all
players that leads to nobody wanting to deviate from that set of actions. The strategy profile is then ‘locked in’.
→ Properties of Nash Equilibria;
- Nash equilibria exibit stability. No player wants to deviate, there is an equilibrium in true sense.
- Nash equilibria always exist. No matter what the payoffs are, there will be at least one equilibrium (mixed or pure).
- Nash equilibria can be sensitive to perturbations. If a player slightly deviates (due to an exogenous reason) from a Nash
equilibrium strategy, the outcome might converge to a wildly different strategy profile.
- Nash equilibria are not necessarily efficient. It can be an inferior outcome of a game (prisoner’s dilemma).
- Nash equilibria do not have to be unique. There can be multiple Nash equilibria, which leads to strategic uncertainty.
- Nash equilibria will never contain dominated strategies.
Example Best response correspondence
We can capture a mixed strategy with one value (probability); player 1 playing A with probability p
and playing B with probability (1-p). Player 2 playing X with probability q and Y with probability (1-q).
→ Find the best response BR2(p, 1 – p) for player 2. Where 𝑞 ∈ [0, 1].
1. Calculate the expected utilities
o What is the expected utility for player 2 playing X; 𝒖2((p, 1 – p), 𝐗) = 4 ⋅ p + 3 ⋅ (1 – p) = 3 + p
o What is the expected utility for player 2 playing Y; 𝒖2((p, 1 – p), 𝐘) = 2 ⋅ p + 4 ⋅ (1 – p) = 4 – 2p.
2. What do these expected utilities tell us?
o Player 2 prefers 𝐗 over 𝐘 if: 𝒖2((p, 1 – p), 𝐗) > 𝒖2((p, 1 – p), 𝐘); so if p > 1/3 and BR1(p, 1-p) = 1
o Player 2 prefers 𝐘 over 𝐗 if: 𝒖2((p, 1 – p), 𝐗) < 𝒖2((p, 1 – p), 𝐘); so if p < 1/3 and BR1(p, 1-p) = 0
o Player 2 is indifferent between 𝐗 and 𝐘 if: 𝒖2((p, 1 – p), 𝐗) = 𝒖2((p, 1 – p), 𝐘); so either choice is correct
3. How about the best response correspondence for player 1, in terms of probability q?
o If q < 2/3, then player 1 chooses B
o If q > 2/3, then player 1 chooses A
o If q = 2/3, then player 1 is indifferent
4. Find the pure strategy Nash equilibria.
o If player 2 plays X, then player 1’s best response is A
o If player 2 plays Y, then player 1’s best response is B
o If player 1 plays A, then player 2’s best response is X
o If player 1 plays B, then player 2’s best response is Y
! payoff pairs that are marked are Nash equilibria, so (A,X) and (B,Y)
5. Find the mixed strategy Nash equilibria.
o The best response correspondence of player 2 was BR2(p, 1 – p)
▪ if p < 1/3, player 2’s BR was Y and player 1’s Best response was B.
▪ if p > 1/3, player 2’s BR was X and player 1’s Best response was A.
▪ In order for player 2 to randomise, player 1 should play 𝝈1* = (1/3, 2/3).
2) The best response correspondence of player 1 was BR1(q, 1 – q)
▪ Player 2 randomises 𝝈2 = (q, 1 - q), in order to make player 1 indifferent between A and B.
▪ 𝑢1(𝐀, (𝑞, 1 – 𝑞)) = 𝑢2(𝐁, (𝑞, 1 – 𝑞) → q = 2(1 – q) → q = 2/3
▪ In order for player 1 to randomise, player 2 should play 𝝈2* = (2/3, 1/3).
3) (𝝈1*, 𝝈2*) = ((1/3, 2/3), (2/3, 1/3)) is therefore a mixed strategy Nash equilibrium.
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