Introductory Game Theory
Week 1: Utility, Decision theory, Rationality
Needed Skills:
be able to solve for X.
Know some calculus.
taking derivative
- power rule basically.
Assignments:
Problem sets: homework assignments due the week after they are handed out.
- you will get stuck.
One exam
after bootcamp
closed note exam
One paper
- equilibrium spaces and comparative statics is important for the paper.
What is Game Theory:
Strategic interaction - two or more players with potentially conflicting goals.
- Players’ choices depend on what others might do.
Interdependence!
Two assumptions:
Common knowledge: something is common knowledge if everyone knows it, everyone knows that everyone knows it.
structure of game is always common knowledge
breaking this assumption gets interesting but is hard.
don’t forget that it is there!
Rationality: players will act to maximize utility
players are goal-oriented
trying to achieve the most value of the game
not making moves arbitrary.
players are trying to accomplish something.
utility is a rank-ordering of outcomes.
actions: rock, paper, scissors
outcomes: win, lose, draw
actors are concerned with the outcome space
- actions are instrumental.
players are playing to win the game.
Players assume all other players are rational.
Models must specify:
actors
strategies (their available choices or actions)
interests (utilities over outcomes)
Utility:
assigns numbers to outcomes in some way
we want to be able to do math on our utility functions.
utility is just a rank ordering
- a utility function is where assign some number to it.
Utility Functions:
Function: a mapping of inputs to outputs
the input is an outcome of the game.
vertical line rule!
- every input must return a unique output.
should work smoothly with respect to lotteries/gambles.
discrete = finite
Expected utility (EU)
preference ranking
averaging across discrete events
used to explore how decisions are made - how we compare choices to one another under uncertainty.
VNM axioms:
Completeness:
For any A,B, either A > B, B > A, or A = B
intuitively: utility value is assigned to everything
Continuity:
- if A > B > C, there exist some \(p \in [0,1]: pA + (1-p)C = B\)
Transitivity:
A >B and B>C, A>C
no preference cycles
Independence:
if A > B, then for any \(p \in [0,1], pA+(1-p)C>pB+(1-p)C\)
preference between A and B should be the same regardless of irrelevant alternative C.
All we are assuming is that actors have goals and we can rank those goals.
- this is rationality.
Thin vs. Thick rationality
thin: just assume goal oriented
you could make Charles Manson rational with this
kinda like overfitting
Thick: more bounded.
- convention
Changing the utility function mapping captures risk-aversion
- can also do risk-acceptance
Convex vs. concave utility
Convex: risk acceptant
opens upward
- “concave up” - Josh
Concave: is risk averse
opens downward
- “concave down” - Josh
Reading (Chapter 1)
Decision Problem has 3 features:
Actions: all the alternatives from which the player can choose.
Outcomes: the possible consequences that can results from any of the actions
Preferences: describe how the player ranks the set of possible outcomes from most desired to least desired. The preference relation describes the player’s preferences and the notation \(x \succsim y\) means “x is at least as good as y.”
Notation
\(x \succsim y\) means “x is at least as good as y.” - preference relation
\(x \succ y\) means “x is strictly better than y.” - Strict preference relation
\(x \thicksim y\) means “x and y are equally as good.” - indifference relation
Assumptions
Players must be able to rank outcomes.
- The completeness Axiom
Transitive axiom.
- at least one best outcome.
Payoff function:
The ability to rank our decisions numerically.
assigning a value to our preferences that can be ranked.
How we actually assign the value is not important
- so long as they represent the correct ordering of our preferences.
Rational Choice Paradigm:
An individual is rational in that he chooses actions that maximize his well-being as defined by his payoff function over the resulting outcomes
This assumption is the rational choice paradigm.
We make assumptions within this paradigm
The player fully understands the decision problem by knowing:
all possible actions; A
all possible outcomes; X
exactly how each action affects which outcome will materialize
his rational preferences (payoffs) over outcomes.
Chapter 2
Decision trees are useful to visualize actions and outcomes.
A= {g,s}; these are the sets of all possible actions
X = {0,10} these represent the outcomes (in profit terms).
We can utilize a decision tree to describe the player’s decision problem that includes uncertainty.
How do players make choices?
they sum over the probabilities of the events they want.
- figure 2.2. Player would choose the action that has the highest probability of achieving the output they desire.
Expected payoff from the lottery
maybe there is a cost associated
figure 2.3
we still average out.
- whichever expected payoff is higher, the actor should choose.
v(a) is used to define the expected payoff of an action
Example:
v(g) = 6.5
v(s) = 5
we read this as the expected payoff of taking action g is 6.5 and the expected payoff of taking action s is 5.
- the player should choose action g as it has the higher expected payoff.
When we choose to use expected payoff then the intensity of preferences matters.
Rational Decision Making with Uncertainty:
We define a rational player as one who chooses an action that maximizes his payoffs among the set of all possible actions.
FIGURE 2.4
One thing im struggling with is how we determine the payoffs and cost.
Applications
- When decisions lead to stochastic outcomes, our rational player chooses his action to maximize his expected payoff so that on average he is making the right choice.
Week 2: Discrete Choice Games in Normal Form
Lecture Notes:
Homework is suppose to be hard.
Turn in hard copy and include your scratch work.
Normal Form aka strategic form
extensive form games are the trees
- more flexible
Prisoner’s Dilemma
keep quiet or talk
games have players, action/outcome, interests.
normal form game
cell entries are outcomes
strategies are talk or quiet
Player 1 always the ROW player
Player 2 always the COLUMN PLAYER
player 1, player 2
Normal form games are always simultaneous
- Utility function - input is the final decisions (the strategy combinations) -> outputs is a cardinal rank order of the strategy combinations.
A solution is a prediction about how game will be played.
- We want to identify ‘reasonable’ ways to play.
We want a systematic rule to produce solutions
we need some standard
any rule that produces solution is a solution concept
josh thinks of solution concepts as gatekeeprs
they eliminate unreasonable strategies rather than identifying reasonable ones.
it is often more informative that something is not a Nash equilibrium than that something is.
Nash Equilibrium
2nd most important concept in this course.
Nash depends on the concept of: best reply or best response
- this is the most important thing.
best response correspondence
- going through every possible strategy by one player and figuring out the other’s best response.
Nash is a set of mutual best replies
in prisoner dilemma - the both defect cell is a nash.
no regret basically
To find Nash - we assume or stipulate a strategy profile.
- if no individual can profitably deviate from that profile, it is Nash.
never compare across diagonals!
Nash is based off rational anticipation.
Nash involves a common conjecture
I play x because I think you’ll play y; I also think it’s reasonable for you to play y given I will play x.
Beliefs: what one player believes the other’s strategy will be.
every game has at least one Nash Equilibrium*
a profile that fails to be Nash does not have a mutually-reinforcing common conjecture.
Nash doesn’t make predictions about which equilibrium actually happens.
Rationtalizable
A strategy is rationalizable if it is a best reply to at least one strategy that can be played by the opponent.
in prisoner dilemma, Cooperate is not rationalizable because it is never a best reply under any conditions.
thought process being modeled.
Chapter 3: Preliminaries
3 elements
possible actions
the deterministic or probabilistic relationship between actions and outcomes
the decision maker’s preferences over the possible outcomes.
We need to understand what other players are going to do.
- what is the strategic environment.- Strategic environment: you have to think hard about what other players are doing in order to decide what is best for you - knowing that the other players are going through the same difficulties.
Static Games:
a player makes a once-and-for-all decision, after which outcomes are realized.
2 steps - these settle what we mean by static:
1) Each player simultaneously and independently chooses an action.
2) Conditional on the players’ choices of actions, payoffs are distributed to each player.
Complete information:
- all players understand the environment they are in - that is, the game they are playing- in every way.
Games of Complete Information
All the possible actions of all the players
all the possible outcomes
how each combination of actions of all players affects which outcome will materialize
the preferences of each and every player over outcomes.
Common knowledge:
- Everyone knows E, everyone knows that everyone knows E.
Normal-form Games with Pure Strategies
A Normal-form game consists of three features:
set of players
- \(N={1,2….,n}\)
set of actions for each player
a collection of sets of pure strategies
- \({S_1, S_2,….,S_n}\)
a set of payoff functions for each player that give a payoff value to each combination of the players’ chosen actions.
Strategy: a plan of action intended to accomplish a specific goal.
A pure strategy for player i is a deterministic plan of action. The set of all pure strategies for player i is denoted \(S_i\).
only focusing on deterministic pure strategy right now. Not stochastic.
Example: The Prisoner’s Dilemma
static game
complete information
best strategy is to confess.
Example: Battle of the Sexes
There is no strategy that is always best.
playing F is best of your opponent plays F.
playing O is best if your opponent plays O.
does not provide a useful solution concept.
Solution Concept:
a method of analyzing games with the objective of restricting the set of all possible outcomes to those that are more reasonable than others.
We will consider some reasonable and consistent assumptions about the behavior and beliefs of players that will divide the space of outcomes into “more likely” and “less likely”
if rules are set, how people are going to react - this is the solution concept.
There are different solution concepts.
Dominant strategy
Rationalization
Nash Equilibrium
Equilibrium:
For any one strategy profiles that emerges as on of the solution concept’s predictions
We often think of equilibria as the actual predictions of our theory.
we add a fourth assumption
self-enforcement.
constrains the set of out outcomes that are reasonable.
- critical assumption for noncooperative game theory
Non-Cooperative game theory:
Each player is in control of his own actions and he will stick to an action only if he finds it to be in his best interest.
If a profile of strategies is to be an equilibrium, we will require each player to be happy with his own choice given how the others make their own choices.
Competitive environments where players act independently and don’t form alliances.
Pareto Optimality:
an outcome is considered to be socially undesirable if there is a different outcome that would make some people better off without harming anyone else.
efficiency, no waste.
kinda seems like another way to say you cannot “iterate” any further? Not sure if this is correct.
Chapter 4: Rationality and Common Knowledge
Dominant Strategy
In a prisoner dilemma, defection strictly dominates cooperation
- There is a dominant strategy
Rational players never play a strictly dominated strategy.
In the Battle of the Sexes game
Neither players has a dominant strategy.
if we stick to the solution concept of strict dominance where there is no equilibrium.
- most games do not have strict dominance.
Iterated Elimination of Strictly Dominated Pure Strategies
A rational player will never play a dominated strategy.
If a rational player has a strictly dominated strategy then he will play it.
Cournot Duopoly
firms simultaneously chooses a quantity of production.
firms want to maximize profit.
firms are in equilibrium when each does not want to change what it is doing given the other firm’s strategy.
Beliefs, Best Response, and Rationalizability
Strict dominance and IESDS are based on eliminating actions that players would never play.
An alt approach: what possible strategies might players choose to play and under what conditions?
Difference between decision theory and game:
- in a game your optimal decision not only depends on the structure of the game, but it will often depend on what the other players are doing.
qualify strategy
In order for a player to be optimizing in a game, he has to choose a best strategy as a response to the strategy of his opponents.
if i think my opponents are doing \(s_{-i}\) , then I should do \(s_i\)
Rationalizability
Instead of asking what your opponents might be doing, you asked
“what would a rational player not do?”
- we are now going to do basically the inverse.
“What might a rational player do?”
- a rational player will select only one strategies that are a best response to some profile of his opponents.
We can eliminate all strategies that are never a best response.
What survives is called the set of rationalizable strategies.
Rationalizability and IESDS are basically two sides of the same coin
- some slight difference is mentioned that will be elaborated later.
Chapter 5: Pinning Down Beliefs: Nash Equilibrium
For dominant strategy equilibrium we required only that players be rational
IESDS and rationalizability we required common knowledge of rationality
Nash Equilibrium: is a system of beliefs and a profile of actions for which each player is playing a best response to his beliefs and, moreover, players have correct beliefs.
best response to a best response.
Requirements for Nash Equilibrium:
Each player is playing a best response to his beleifs
the beliefs of the players about their opponents are correct.
does not guarantee pareto optimality.
self-enforcing outcomes.
left to their own devices, people in many situations will do what is best for them, at the expense of social efficiency.
Citation
@online{neilon2025,
author = {Neilon, Stone},
title = {Introductory {Game} {Theory}},
date = {2025-01-14},
url = {https://stoneneilon.github.io/notes/Game_Theory/},
langid = {en}
}