This document contains all the notes from the synchronous and asynchronous lectures throughout the year and is split up lecture by lecture. This document on its own, is sufficient to get a top mark in your end of year exam for IB3H9 Strategic Games Notes.
Strategic Games: Thinking rationally about business, policy and real life
Game theory is the study of strategic interaction. Two or more people act – independently
and in their own self-interest – and then the outcomes depend on the decisions that each of
them make. Interaction with other equally rational decision makers.
The Beauty Contest Guessing Game: John Maynard Keynes’
The game shows how the market is a guessing game: To estimate the value of a stock now,
we need to estimate what others will pay for the stock in the future. They are estimating its
value based on what we will pay for it in the future
Keynes likened this to a special kind of beauty contest: Professional investment may be
likened to those newspaper competitions in which the competitors have to pick out the 6
prettiest faces from a hundred photographs, the prize being awarded to the competitor
whose choice most nearly corresponds to the average preferences of the competitors as a
whole. This game is not a case of choosing those which, to the best of ones judgement are
the prettiest. We have reached the third degree, where we devote our intelligences to
anticipate what average opinion expects the average to be.
For this game, we assume everyone knows that everyone is rational, and that everyone
knows that everyone knows that everyone is rational. This is common knowledge of
rationality.
If everyone is fully rational, and assumes everyone else is fully rational, they will end when
the adjustment process leads to an unchanging strategy on both sides (0). 0 is the Nash
equilibrium.
In practise, however, very few people select 0. Most around 20-30
Strategic Situations arise whenever there is interdependence between actors. What is best
for each depends on what others do.
It is called “game theory” as: there are rules of play; actions each player can take; payoffs –
a relationship between actions and outcomes; no actor individually determines the
outcome. Strategic interdependencies: you gain an advantage by considering the responses
of others to your actions, and they will be doing the same
Equilibrium: An equilibrium is a conjunction of strategies, one for each actor, from which no
actor will want to move, if no other actor moves at the same time.
Hotelling’s law: “there is an undue tendency for competitors to imitate each other in quality
of goods, in location, and in other essential ways”
Representing games: A set of players: how many they are
Actions for each player: available strategies, what can the players do
Each player’s preferences about the action profile: preferences over the list of all the
players’ actions.
What do payoffs represent: The payoffs represent what the outcomes are worth to the
players.
,Ordinal utility: Larger numbers are better. But nothing can be said about differences. Simply
a rank.
Cardinal utility: An interval-level measure. Differences between numbers can be compared.
Not simply a rank, differences matter
Utilities are not interpersonally comparable – utilities cannot be added up or compared
across players
Modelling Games:
Two basic types of strategic interactions: Simultaneous and Sequential
Simultaneous games: Time is absent from the model; Players act at the same time; No
player is informed, when she chooses her action, of the action chosen by any other player.
Sequential games: Timing plays a role; Players make alternate moves; The later players have
some information of the first player’s choice
Games can be modelled in two ways:
Normal form: Use a matrix; Usually describe simultaneous games
Extensive form: Use a game-tree; Usually describe sequential games. In Extensive form, the
payoff which is written first is the payoffs of the first mover. Decision nodes is the points at
which each player has to take an action.
Game trees represent differences in the information available to the actors.
Two crucial elements: Timing; Information
The second player knows what the first player has chosen before they choose themselves
but the first person also knows that their action will be observed.
Information sets: In this game tree, there is an oval drawn around the two
choice nodes of Harry. This means that when Harry chooses, he does not
know what Sally has chosen. This means the game is not sequential, but
simultaneous. (Or, he does not know at which decision node he is. The oval
circles Harry’s Information Set
This is equivalent to a 2X2 game in a Normal form because neither player knows what the
other has done.
Solving sequential games: Look forward, reason backward:
You say: what will the final player do? Then what will the
next to final player do, given what they know the final
player will do? This is rollback or backward induction.
You have to go via node by node and only keep the best
option at each node
The path we end up on is called the equilibrium path
A strategy is a complete conditional plan for a player in the
game. Complete conditional: describes what she will do at each of her information sets.
Information set is each node. If you do the number of strategies/options to the power of
information sets you get the number of strategies for the individual player.
, Writing strategies for a player i: Find every information set for player i; At each information
set, find all actions; Find all combinations of actions at these information sets.
For example, the strategies available to player 1 is just: Y, N. These are all the strategies
available to player 1.
The strategies available to player 2 is: YY, YN, NY, NN. These are only the strategies available
to player 2. Example YN means that at the second node, the second player will choose “Y”,
and at node c, the second player will choose N
Strategies available to player 3: YYYY, YYYN, YYNY, YYNN, YNYY, YNYN, YNNY, YNNN, NYYY,
NYYN, NYNY, NYNN, NNYY, NNYN, NNNY, NNNN
YYNY means Y at node d, Y at node e, N at node f, and Y at node g.
The rollback equilibrium: (N, NY, NYYN). The equilibrium path: NYY.
What if there were 650 sequential votes? We can still use rollback, but we can’t easily draw
the tree: MP 650: Yes if exactly 325 Yes votes already. MP 649: Yes if exactly 324 Yes votes…
MP 325: Yes if 324 No votes, Otherwise No. MP 324: No… MP 1: No
Ultimatum Game Reponses: FMRI responses (Sanfey et al., 2003):
Players who received unfair offers had significantly greater activation in several areas of
their brain compared to those who received fair offers: the right dorsal lateral prefrontal
cortex (associated with planning); anterior insula (associated with pain and disgust) and the
anterior cingulate cortex (associated with resolving conflicts among different regions of the
brain).
The Nash Equilibrium: A strategy is a best response to a particular strategy of another
player, if it gives the highest payoff against that particular strategy. How to find best
responses: for each of opponent’s strategy, find the strategy yielding best payoff.
A set of strategies form a Nash equilibrium if and only if the strategies are best responses to
each other. This means given all players follow Nash, no individual player would want to
deviate.
More formally: Two components: Each player chooses their action rationally, given their
beliefs about the other players’ actions; Every player’s belief about the other players’
actions is correct.
¿
A Nash equilibrium is an action profile ∝ with the property that no player i can do better by
¿ ¿
choosing an action different from ∝i , given that every other player j adheres to ∝ j .
Ideally it is a steady state: if everyone else adheres to it, no one wishes to deviate.
If other players knew your strategy and would not want to change theirs, and if you knew
other players strategy and would not want to change yours, you are at a Nash equilibrium
Nash equilibria are not necessarily optima… either for the players or for society
Optimality: Players can be better off; Society can be better off; Rationality can lead to
socially destructive outcomes.
An analysis of Thomas Schelling: Micromotives: what each person prefers
Macrobehaviour: The aggregate effect of choices based on these preferences. The
aggregate effects of people acting in their own best interest in response to the choices of
others are often not desired nor anticipated by the individual actors.
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