ECON 3M03 – Game Theory – Exam Review
Exam Coverage
- Cumulative
- Pre-Midterm:
- Post-Midterm:
(21 total chapters)
- Chapters 1 – 5, 6 – 8, 9 – 11
(11 total chapters)
- Chapters 12, 14 – 17, 22 – 23, (24, 26, 27) (10 total chapters)
Chapters 1-5: Representing Games
Module (Chapters 1-5) Outline
- Game Theory: Definitions and overview
- Representation: Extensive forms
- Strategies
- Representation: Normal forms
- Beliefs, Mixed Strategies, and Expected Payoffs
- General Assumptions and Methodology
Chapter 1: Introduction
What is a Game?
- A game is being played whenever people interact with each other
- Chess, poker, tennis, etc.
- Bidding in an auction, pricing at Amazon.com, etc.
- Firm competition, international relations, firm-employee relations, etc.
- Interdependence: one person’s behavior affects another’s well-being
What is NOT a Game?
- When 𝑁 = 1: monopoly
- Or when 𝑁 = ∞: perfect competition
Three Major Tensions of Strategic Interaction
- Game Theory: a theory of strategic interaction
- Conflict
- Cooperation
- Three major tensions:
- Conflict between individual and group interests
- Strategic Uncertainty: not knowing for sure what other players will do
- Inefficient Coordination
Noncooperative vs. Cooperative Game Theory
- Noncooperative Game Theory:
- Examine individual decision making in strategic settings
- Does not rule out Group Decision Making: need to specify procedures leading
individual decisions to group outcomes
- Develop Solution Concepts: predictions about the outcomes of games
- * We focus on noncooperative game theory in this course *
- Cooperative Game Theory:
- Model Joint Actions: parties negotiate and jointly agree on the terms of their
relationship
Representing Games: Key Elements
- A list of players
- A complete description of what players can do
- A description of what the players know when they act
- A specification of how the players’ actions lead to outcomes
- A specification of the players’ preferences over outcomes
Chapter 2: Extensive Form
Extensive Form and Normal Form
- Two basic types of interactions:
- Sequential: players make alternating moves
- Simultaneous: players act at the same time
- In most cases, interactions are partly sequential and partly simultaneous
- Noncooperative games can be modeled in two ways:
- Extensive Form: ie. game tree
- Normal Form: ie. payoff matrix
Extensive Form (Game Trees)
- A Game Tree consists of:
- A series of nodes linked in a sequence
- Nodes are where things happen
- Non-Terminal Node: not an endpoint
- Terminal Node: indicates that the game is over
- Branches represent individual actions taken by the players
- * Please Note: loops (ie. cycles) are not allowed in game trees *
- Two Crucial Elements of Extensive Form Games:
1. Timing of actions that players may take
2. Information players have when they must take those actions
- Information Sets: summarize a player’s knowledge of prior moves when he
must decide
- Example: The Bug Game
- In 1998, two movies were in theatre and both were about bugs: A Bug’s Life (Disney)
and Antz (DreamWorks). Rumor has it that Jeffrey Katzenberg, former Disney CEO,
resigned from Disney and later joined DreamWorks and stole the bug movie concept
from Disney
- A tale of two films:
- Disney: A Bug’s Life
- DreamWorks: Antz
- A model:
- Set of players:
- Jeffrey Katzenberg (player K)
- Michael Eisner (player E)
- Set of actions for each player, etc.
- Building an Extensive Form, Step-by-Step:
- The game tree is defined by nodes and branches:
- Nodes: are solid circles that represent places where a decision is made by one
of the players
- Branches: are arrows connecting the nodes. They indicate the various actions
that players can choose
- Katzenberg’s first move:
- Adding the production decisions:
- Capturing lack of information:
- Note: when K is on the move at either c or d, he knows that he is at one
of the two nodes but he does not know which one. K cannot distinguish
between nodes c and d
- The dashed line connecting c and d illustrates this lack of information
- Information Sets:
- An information set is a place where a decision is made
- Information sets summarize a player’ knowledge of prior moves when
he must decide
- If there are more than one nodes in an information set, a player knows
that she is in one of the nodes in the information set (but does not know
which one)
- Information sets containing only one node are referred to as singletons
- Adding terminal nodes:
- The information set for a comprises just this node since K can
distinguish this node from his other nodes
- c and d are in the same information set
- b and e are their own separate information sets
- Only one decision is made at each information set. An information set
describes which nodes are connected to each other by dashed lines
- Altogether, player K has 3 decisions; player E has only one decision
- Decision Nodes: a, b, c, d, e; Terminal Nodes: f, g, h, l, m, n
- Add payoffs:
- The Full Extensive Form:
- A more compact representation:
- Labelling Branches:
- Differentiate between N and N’: do not label two nodes for the
same player with the same actions
- Conformity within an information set
- More Examples:
- Price-Competition Game:
(a) - Player 1 has one information set; player 2 has two information sets
- Game tree (a) depicts the same game as (c), except that the players’ payoffs
are different
(b) - The two firms select their prices simultaneously and independently. Player 2
doesn’t get to observe player 1’s selection before making its own choice.
Player cannot distinguish between its two decision nodes: thus, we connect
them using a dashed line
- Player 1 has one information set; player 2 has one information set
- Exercise: Represent the following game in extensive form:
- Firm A decides whether to enter firm B’s industry. B observes this decision. If A enters,
then the two simultaneously decide whether to advertise. Otherwise, B alone decides
whether to advertise
- With two firms in the market, the firms earn profits of $3 million each if they both
advertise and $5 million if they both do not advertise
- If only one firm advertises, then it earns $6 million and the other earns $1 million
- When B is solely in the industry, it earns $4 million if it advertises and $3.5 million if it
doesn’t advertise
- Firm A earns nothing if it does not enter
- Solution:
Chapter 3: Strategies and Normal Form
Strategy
- Strategy: is a complete contingent plan for a player in the game
- It is a full specification of a player’s behavior, which describes what a player will do at
each of their information sets
- A strategy implies a path through the tree, leading to a terminal node and payoff
vector
- Steps: Writing strategies for a player 𝑖
1. Find every information set for player 𝑖
2. At each information set, find all actions
3. Find all combinations of actions at these information sets
- Strategy: Terminology and Notation:
- A strategy of player 𝑖 is denoted as 𝒔𝒊
- A Strategy Set (or strategy space) 𝑺𝒊 for player 𝑖 is the set of all possible strategies
available to player 𝑖
- Individual Strategy: 𝒔𝒊 ∈ 𝑺𝒊
- ie. 𝑺𝒊 = {𝐻, 𝐿} and 𝒔𝒊 = 𝐻
- Strategy Profile: is a vector of strategies, one for each player. It’s denoted 𝒔. So with 𝑛
players, we have 𝒔 = (𝒔𝟏 , 𝒔𝟐 , … , 𝒔𝒏 ), where 𝒔𝒊 is the strategy of player 𝑖
- The set of all possible strategy profiles is denoted 𝑺:
-𝒔∈𝑺
- Since strategies are just contingent plans of action (which may or may not be
good plans), one player having some particular strategy never rules out another
player having any particular strategy
- This means that with 𝑛 players, 𝑺 = 𝑺𝟏 × 𝑺𝟐 × … × 𝑺𝒏
- The symbol “×” denotes the Cartesian product or cross product of sets
- ie. Assume 𝑆1 = (𝐴, 𝐵), 𝑆2 = (𝑋, 𝑌) → 𝑆 = 𝑆1 × 𝑆2 =
{(𝐴, 𝑋), (𝐴, 𝑌), (𝐵, 𝑋), (𝐵, 𝑌)}
- 𝒔−𝒊 is a strategy profile involving every player except player 𝑖
- Here “−𝑖” means player 𝑖’s opponents
- 𝒔−𝒊 = (𝒔𝟏 , 𝒔𝟐 , … , 𝒔𝒊−𝟏 , 𝒔𝒊+𝟏 , … , 𝒔𝒏 )
- 𝑺−𝟏 is the set of all such incomplete strategy profiles
- Example: Exit Decisions:
1. Find the number of information sets for Players 1 and 2
- The game has one information set for firm 1 and one for firm 2
2. Write down the strategy set for each player
- Strategy:
- Firm 1: Aggressive (A), Passive (P), or Out (O)
- Firm 2: Aggressive (A) or Passive (P)
- Strategy Sets:
- Firm 1: 𝑆1 = {𝐴, 𝑃, 𝑂}
- Firm 2: 𝑆2 = {𝐴, 𝑃}
3. Write down the set of strategy profiles
- Set of Strategy Profiles:
𝑆 = 𝑆1 × 𝑆2 = {(𝐴, 𝐴), (𝐴, 𝑃), (𝑃, 𝐴), (𝑃, 𝑃), (𝑂, 𝐴), (𝑂, 𝑃)}
- Exercise: Finding Strategies:
- In this game (centipede game), players decide between “out” (O) and “in” (I)
1. Find the number of information sets for Players 1 and 2
2. Find the number of actions at each information set
3. Write down the strategy set for each player
- Solution:
- Player 1 has 2 information sets. His strategy must specify what he will
do at both of his information sets
- Player 1 has 2 actions at the first information set and 2 actions at
the second information set
- Player 2 only has 1 information set
- Player 2 has 2 actions at their only information set
- 𝑆1 = {𝑂𝐴 , 𝑂𝐵 , 𝐼𝐴 , 𝐼𝐵 }
- 𝑆2 = {𝑂, 𝐼}
- * Attention: *
- The definition of a strategy (a complete contingent plan) requires a
specification of player 1’s choice at his second information set even in the
situation in which he plans to select Out (O) at his first information set
- We have to keep track of behavior at all information sets – even those
that would be unreached if players follow their strategies – to fully
analyze any game
- More Exercises: Find Strategy Sets:
- Player 1 has 1 information set; player 2 has 2 information sets; player 3 has 2
information sets
- 𝑆1 = {𝑈, 𝐷}
- 𝑆2 = {𝐴𝐶, 𝐴𝐸, 𝐵𝐶, 𝐵𝐸}
- 𝑆3 = {𝑅𝑃, 𝑅𝑄, 𝑇𝑃, 𝑇𝑄}
- Player 1 has 2 information sets; player 2 has 1 information set
- Strategy Sets:
- 𝑆1 = {𝐴𝑊, 𝐵𝑊, 𝐶𝑊, 𝐴𝑍, 𝐵𝑍, 𝐶𝑍}
- 𝑆2 = {𝑋, 𝑌}
- The set of strategy profiles:
- 𝑺 = {(𝐴𝑊, 𝑋), (𝐴𝑊, 𝑌), (𝐵𝑊, 𝑋), (𝐵𝑊, 𝑌), (𝐶𝑊, 𝑋), (𝐶𝑊, 𝑌), (𝐴𝑍, 𝑋), (𝐴𝑍, 𝑌),
(𝐵𝑍, 𝑋), (𝐵𝑍, 𝑌), (𝐶𝑍, 𝑋), (𝐶𝑍, 𝑌)}
Normal Form (or Strategic Form)
- A game in normal form consists of:
- A set of players, {1,2, … , 𝑛}
- Strategy spaces for the players, 𝑆1 , 𝑆2 , … , 𝑆𝑛
- Payoff functions for the players, 𝑢1 , 𝑢2 , … , 𝑢𝑛
- 𝑢𝑖 : 𝑺 → 𝑹, (a function whose domain is the set of strategy profiles and whose
range is
the real numbers)
- For each strategy profile 𝒔 ∈ 𝑺, 𝑢𝑖 (𝑠) is player 𝑖’s payoff in the game
- Compared to the extensive form, normal form can be:
- More compact
- For each extensive form, there exists an equivalent normal form representation
Classic Normal Form Games
- Example: Prisoners’ Dilemma:
- Set of players: 𝑁 = {Ginger, Rocky}
- Timing: simultaneous moves
- Set of strategies: 𝑆𝑖 = {Confess, Not Confess}
- Set of payoffs:
- If one confesses, the other does not: the payoff is 0 and 15 years in jail,
respectively
- If both confess: each gets 5 years in jail
- If neither confess: each gets 1 year in jail
Rocky
Confess
Not Confess
Confess
–5, –5
0, –15
Not Confess
–15, 0
–1, –1
Ginger
- Find strategy sets and strategy profiles:
- For Ginger: 𝑠𝐺 = 𝐶 or 𝑠𝐺 = 𝑁 so 𝑆𝐺 = {𝐶, 𝑁}
- Similarly, for Rocky: 𝑠𝑅 = 𝐶 or 𝑠𝑅 = 𝑁 so 𝑆𝑅 = {𝐶, 𝑁}
- This gives us four possible strategy profiles: 𝑠 = (𝐶, 𝐶), 𝑠 = (𝐶, 𝑁), 𝑠 =
(𝑁, 𝐶),
or 𝑠 = (𝑁, 𝑁)
- Strategy Profiles: 𝑺 = {(𝑪, 𝑪), (𝑪, 𝑵), (𝑵, 𝑪), (𝑵, 𝑵)}
- Classical Game: Matching Pennies:
- Zero-Sum Game: sum of payoffs in each cell is zero
- Classical Game: Coordination:
- Coordination: want to use the same strategy, (𝐴, 𝐴) or (𝐵, 𝐵) ie. traffic rules
- Classic Game: Pareto Coordination:
- Coordination: want to select the same strategy
- Pareto Coordination: prefer to coordinate on A rather than on B
- Classic Game: Battle of the Sexes:
- Coordination Game: want to go to an event together (ie. coordinate) but with slightly
different preferences
- Classic Game: Chicken aka Hawk-Dove:
- Two players drive cars toward each other at top speed. Just before they reach each
other, each chooses between maintaining course (H) and swerving (D)
Corresponding Extensive and Normal Forms
- Both extensive forms yield the same normal form
- * Note: *
- One way of viewing the normal form is that it models a situation in which players
simultaneously and independently select complete contingent plans for an extensive
form game
- This demonstrates that although there may be only one way of going from extensive
form to the normal form, the reverse is not true
Exercise: The Katzenberg-Eisner Game:
1. Describe the strategy spaces (discussed earlier)
2. Draw the normal-form representation
- Solution:
- K has 3 information sets. A strategy for K is a combination of 3 actions, one for
each of his information sets, ie. LNR
- K has two choices at each of the three information sets, so there are
2 × 2 × 2 = 8 different combinations
- Player K’s strategy space: 𝑺𝑲 =
{𝑳𝑷𝑹, 𝑳𝑷𝑵′ , 𝑳𝑵𝑹, 𝑳𝑵𝑵′ , 𝑺𝑷𝑹, 𝑺𝑷𝑵′ , 𝑺𝑵𝑹, 𝑺𝑵𝑵′ }
- E has 1 information set
- E’s strategy space: 𝑺𝑬 = {𝑷, 𝑵}
- * Note: * It must have 8 rows for the 8 different
strategies of player K and 2 columns for the 2 strategies of
player E
- For each individual strategy profile, we can trace through
the game tree to find the payoff
Chapter 4: Beliefs, Mixed Strategies, and Expected Payoffs
Beliefs
- A player’s assessment about the strategies of the others in the game
- Example of belief in words: Player 1 might say, “I think player 2 is very likely to play
strategy L”
- Representing beliefs:
- The statement “very likely to play L” is ambiguous
- Translate into probability numbers
- Normal form games: probability distribution over the strategies of the other players
Beliefs for General Normal Form Games: Terminology and Notation
- A belief of player 𝒊 is a probability distribution over the strategies of the other players
- 𝜽−𝒊 ∈ ∆𝑺−𝒊 , where ∆𝑺−𝒊 is the set of probability distributions over the strategies of all players
except player 𝑖
- ie. in a two-player game (ie. −𝑖 = 𝑗), and assume each player has a finite number of
strategies:
- The belief of player 𝑖 about the behaviour of player 𝑗 is a function 𝜽𝒋 ∈ ∆𝑺𝒋 such
that, for each strategy 𝒔𝒋 ∈ 𝑺𝒋 of player 𝑗, 𝜽𝒋 (𝒔𝒋 ) is interpreted as the probability
that player 𝒊 thinks player 𝒋 will play 𝒔𝒋
𝜽𝒋 (𝒔𝒋 ) ≥ 𝟎∀𝒔𝒋 ∈ 𝑺𝒋
∑ 𝜽𝒋 (𝒔𝒋 ) = 𝟏
𝒔𝒋 ∈𝑺𝒋
- A belief does not have to be sensible, intelligent, well-founded, or justifiable. It
just has to be a probability distribution over the other players’ strategies
- It may reflect uncertainty on the part of the player holding the belief
Review: Probability
- Probability: likelihood that a given outcome will occur
- Subjective Probability: is the perception that an outcome will occur
- Expected Value: probability-weighted average of the payoffs associated with all possible
outcomes
- Payoff: value associated with a possible outcome
- The expected value measures the central tendency – the payoff or value that we would expect
on average
- ie. Expected Value = Pr (success)($40/share) + Pr (failure)($20/share)
= (1/4)($40/share) + (3/4)($20/share) = $25/share
- With two possible outcomes, the expected value is: 𝑬(𝑿) = 𝑷𝒓𝟏 𝑿𝟏 + 𝑷𝒓𝟐 𝑿𝟐
- When there are 𝑛 possible outcomes, the expected value becomes:
𝑬(𝑿) = 𝑷𝒓𝟏 𝑿𝟏 + 𝑷𝒓𝟐 𝑿𝟐 + ⋯ + 𝑷𝒓𝒏 𝑿𝒏
- Example: Prisoners’ Dilemma:
Rocky
Ginger
0.25
0.75
Confess
Not Confess
Confess
–5, –5
0, –15
Not Confess
–15, 0
–1, –1
- Ginger’s expected payoff from “Confess” = 0.25(−5) + (0.75)(0) = −1.25
Rocky
Ginger
Confess
Not Confess
0.25
Confess
–5, –5
0, –15
0.75
Not Confess
–15, 0
–1, –1
- Rocky’s expected payoff from “Not Confess” = 0.25(−15) + (0.75)(−1) = −4.5
Mixed Strategy
- Pure Strategies: strategy in which a player makes a specific choice or takes a specific action
- Mixed Strategy: is a probability distribution over pure strategies:
- We denote a mixed strategy for player 𝑖 as 𝝈𝒊 ∈ ∆𝑺𝒊
- It is a function mapping a set of possible strategies (𝑺𝒊 ) into [0, 1] such that the sum of
probabilities associated with all possible strategies is 1
- When player 𝑖 has a belief 𝜽−𝒊 about the strategies of the others and plans to select
strategy 𝒔𝒊 , his expected payoff is the weighted average that he would get if he played
strategy 𝒔𝒊 and the others played according to 𝜽−𝒊
- Example: Prisoners’ Dilemma:
𝟏 𝟐
- A mixed strategy 𝝈𝑮𝒊𝒏𝒈𝒆𝒓 = (𝟑 , 𝟑) means Ginger will choose to “confess” with
probability 1/3 and “not confess” with probability 2/3
- The set of mixed strategies includes the set of pure strategies:
- 𝝈𝑮𝒊𝒏𝒈𝒆𝒓 = (𝟏, 𝟎) means Ginger will choose to “confess” with probability 1
- Practice Question:
- For the normal form game, assume:
- Player 2 believes that player 1 will select U, C, D with probability ½, ¼, and ¼,
respectively
- Player 2 plans to randomize by picking M and R each with probability ½
- Find player 2’s expected payoff
- Solution:
- First, let’s look at strategy profile (U, M):
- (U, M) occurs when player 1 selects U and player 2 selects M
- Player 1 selects U with probability ½ and player 2 selects M with
probability ½
1 1
1
- Strategy profile (U, M) occurs with probability 2 ∙ 2 = 4
- When (U, M) occurs, player 2 gets payoff 2. So the probability
1
that player 2 gets this payoff is 4
- Based on player 2’s belief, 6 strategy profiles occur with positive
probability: (U,M), (U,R), (C,M), (C,R), (D,M), and (D,R). The six probability
numbers sum to 1
Strategy Profile
Probability
Player 2’s Payoff
(U,M)
1 1 1
∙ =
2 2 4
2
(U,R)
1 1 1
∙ =
2 2 4
0
(C,M)
1 1 1
∙ =
4 2 8
2
(C,R)
1 1 1
∙ =
4 2 8
0
(D,M)
1 1 1
∙ =
4 2 8
3
(D,R)
1 1 1
∙ =
4 2 8
1
1
1
1
1
1
1
- Player 2’s Expected Payoff: 4 ∙ 2 + 4 ∙ 0 + 8 ∙ 2 + 8 ∙ 0 + 8 ∙ 3 + 8 ∙ 1 =
1.25
Chapter 5: General Assumptions and Methodology
- There is a trade-off between simplicity and realism in formulating a game-theoretic model
- Rationality: assume a player will select the action that leads to the outcome he most prefers
ie. maximizing one’s expected payoff
- Rationality does not necessarily imply that the players seek to maximize their own
monetary gains – other factors: altruism, fairness, envy, greed, friendship, etc.
- Common Knowledge: a particular fact F is said to be common knowledge between players if:
- Each player knows F
- Each player knows that the others know F
- Each player knows that every other player knows that each player knows F
- Example:
- Imagine players gathered around a table where F is displayed. Each one can
verify that the others observe F and also can verify the same about everyone
else
- In this example, F is common knowledge between the players
- In conventional analysis of games, the game is common knowledge between the
players
- However, common knowledge of the game does not imply that, during the play
of the game, players have common knowledge of where they are in the
extensive form (ie. information sets comprising multiple nodes)
Chapters 6-8: Best Response and Rationalizability
Module (Chapters 6-8) Outline
- Dominance and Best Response:
- Dominance
- Best Response
- Dominant Strategy Equilibrium
- Rationalizability and Iterated Dominance:
- Dominance-Solvable Equilibrium
Chapter 6: Dominance and Best Response
Dominance
- A pure strategy is dominated if there is a different strategy (pure or mixed) that does better no
matter what the rivals do
- Definition: A pure strategy 𝑠𝑖 of player 𝑖 is dominated if there is a strategy (pure or mixed)
𝜎𝑖 ∈ ∆𝑆𝑖 such that 𝑢𝑖 (𝜎𝑖 , 𝑠−𝑖 ) > 𝑢𝑖 (𝑠𝑖 , 𝑠−𝑖 ), for all strategy profiles 𝑠−𝑖 ∈ 𝑆−𝑖 of the other
players
- It is also called “strict dominance” to emphasize the strict inequality in the definition
- Definition: Weak Dominance: we say that mixed strategy 𝜎𝑖 weakly dominates pure strategy
′
′
) > 𝑢𝑖 (𝑠𝑖 , 𝑠−𝑖
) for at least one
𝑠𝑖 if 𝑢𝑖 (𝜎𝑖 , 𝑠−𝑖 ) ≥ 𝑢_𝑖(𝑠𝑖 , 𝑠−𝑖 ) for all 𝑠−𝑖 ∈ 𝑆−𝑖 and 𝑢𝑖 (𝜎𝑖 , 𝑠−𝑖
′
strategy 𝑠−𝑖 ∈ 𝑆−𝑖 of the other players
- For Player 2: M weakly dominates L. Strategy L is called a weakly dominated strategy
- Example: a pure strategy dominated by a pure strategy:
- For both players, “Not Confess” is strictly dominated by “Confess” (or, we say that
“Confess” strictly dominates “Not Confess”)
- Practice Question Example: is there any strictly dominated strategy?
- Answer: For player 1, U strictly dominates D
- Practice Question: (1) which strategy is strictly dominated for player 1? (2) what about player
2?
- Answer: (1) player 1: M strictly dominates D, U weakly dominates D
(2) player 2: C weakly dominates R
- Example: a pure strategy dominated by a mixed strategy:
- For player 1, is D dominated by a pure strategy?
- Is D dominated by a mixed strategy?
½
½
0
- Answer:
- D is not dominated by any pure strategy
- A mixed strategy (½, ½, 0) dominates D, where player 1 selects U with
probability ½, M with ½, and D with 0
- If player 2 selects L, player 1’s expected payoff:
1
1
= ( ∗ 4) + ( ∗ 0) + (0 ∗ 1) = 2 > 1
2
2
- If player 2 selects R, player 1’s expected payoff:
1
1
= ( ∗ 0) + ( ∗ 4) + (0 ∗ 1) = 2 > 1
2
2
- There are other mixed strategies that dominate D. For example,
2 1
3 2
2 3
1 1 1
(3 , 3 , 0) , (5 , 5 , 0) , (5 , 5 , 0) , and (3 , 3 , 3)
Steps: How to Search for Dominated Strategies
- Step 1: Check whether a strategy is dominated by another pure strategy
- Step 2: If a strategy is not dominated by another pure strategy, then determine whether it is
dominated by a mixed strategy
1. Look for alternating patterns of large and small numbers in the payoff matrix (this will
help you find strategies that may be assigned positive probability by a dominating mixed
strategy)
2. You need to find only one strategy that dominates it
3. Make sure you check the correct payoff numbers
- Practice Question:
a) For player 1, is A dominated by a pure strategy?
b) Is A dominated by a mixed strategy?
- Note: Player 2’s payoff is illustrated by * since we don’t use them
0
P
1-P
- Answer:
a) In this game, A is not dominated by any pure strategy
b) If A is dominated by a mixed strategy, this mixed strategy must assign positive
probability only to B and C, not A. Why?
- Assume this mixed strategy is 𝝈 and it assigns probability 𝒑 to B and (1–
𝒑) to C
- For this mixed strategy 𝝈 to dominate A, we must have:
𝑢1 (𝐴, 𝑋) < 𝑢1 (𝜎, 𝑋)
1 < (0)𝑝 + 4(1 − 𝑝)
3
𝑝<4
𝑢1 (𝐴, 𝑌) < 𝑢1 (𝜎, 𝑌)
2 < 3𝑝 + 1(1 − 𝑝)
1
𝑝>2
𝑢1 (𝐴, 𝑍) < 𝑢1 (𝜎, 𝑍)
3 < 3𝑝 + 4(1 − 𝑝)
𝑝<1
- Any 𝑝 that satisfies all three conditions gives us a mixed strategy that
1
3
dominates A. ie. any 𝑝 where 2 < 𝑝 < 4
- Note: if it weren’t possible to have a 𝑝 satisfying all three conditions,
then A would not be dominated
Solving a Strategic Form Game: Best Response (Reply)
- A strategy is a best response (reply) to a particular strategy of another player if it gives the
highest payoff against that particular strategy
- Definition: Suppose player 𝑖 has belief 𝜃−𝑖 ∈ ∆𝑆−𝑖 about the strategies played by the other
players. Player 𝑖’s strategy 𝑠𝑖 ∈ 𝑆𝑖 is a best response if 𝑢𝑖 (𝑠𝑖 , 𝜃−𝑖 ) ≥ 𝑢𝑖 (𝑠𝑖′ , 𝜃−𝑖 ) for every 𝑠𝑖′ ∈ 𝑆𝑖
- For any belief 𝜽−𝒊 of player 𝑖, we denote the set of best response by 𝑩𝑹𝒊 (𝜽−𝒊 )
How to Find Best Responses
- Discrete Strategy Space: for each of opponent’s strategy, find strategy yielding best payoff
- Continuous Strategy Space: use calculus
- Note:
- There may be more than one best response to a given belief
- In a finite game, every belief has at least one best response
Dominance and Best Response Compared
- Dominance and Best Response: For a given game, let 𝑈𝐷𝑖 be the set of strategies for player 𝑖
that are not strictly dominated. Let 𝐵𝑖 be the set of strategies for player 𝑖 that are best
responses, over all the possible beliefs of 𝑖. Mathematically, 𝐵𝑖 = {𝑠𝑖 |there is a belief 𝜃−𝑖 ∈
∆𝑆−𝑖 such that 𝑠𝑖 ∈ 𝐵𝑅𝑖 (𝜃−𝑖 )}
- ie. if a strategy 𝑠𝑖 is a best response to some possible belief of player 𝑖, then 𝑠𝑖 is
contained in 𝐵𝑖
- In a finite two-player game, 𝐵1 = 𝑈𝐷1 and 𝐵2 = 𝑈𝐷2
- Procedure for calculating 𝑩𝒊 = 𝑼𝑫𝒊 :
1. Look for strategies that are best responses to the simplest beliefs – those beliefs that
put all probability on just one of the other player’s strategies. These best responses are
obviously in the set 𝐵𝑖 so they are also in 𝑈𝐷𝑖
2. Look for strategies that are dominated by other pure strategies; these dominated
strategies are not in 𝑈𝐷𝑖 and thus they are also not in 𝐵𝑖
3. Test each remaining strategy to see if it is dominated by a mixed strategy
- Best Response: Prisoners’ Dilemma:
𝐵𝐺𝑖𝑛𝑔𝑒𝑟 = 𝑈𝐷𝐺𝑖𝑛𝑔𝑒𝑟 = {Confess}
𝐵𝑅𝑜𝑐𝑘𝑦 = 𝑈𝐷𝑅𝑜𝑐𝑘𝑦 = {Confess}
- Best Response: Battle of the Sexes:
𝐵1 = 𝑈𝐷1 = {𝐴, 𝐵}
𝐵2 = 𝑈𝐷2 = {𝐴, 𝐵}
- Practice: For player 1, calculating 𝑩𝒊 = 𝑼𝑫𝒊 :
0
P
1-P
0
- Step 1: Find strategies that are best response to the simplest beliefs
- Against L, player 1’s best response is Y
- Against R, player 1’s best response is X
→ 𝑋 ∈ 𝐵1 and 𝑌 ∈ 𝐵1
- Step 2: Check whether any strategy is dominated by another pure strategy
- W is dominated by Z
→ 𝑊 ∉ 𝐵1
- Step 3: Test the remaining strategy Z to see if it is dominated by a mixed strategy
- If a mixed strategy were to dominate Z, it would be a mixture of X and Y
- Assume X is played with probability 𝑝 and Y with 1 − 𝑝
- If Z is dominated, then we must have:
2𝑝 + 6(1 − 𝑝) > 5
1
𝑝<4
7𝑝 + 1(1 − 𝑝) > 3
1
𝑝>3
- Impossible to simultaneously satisfy both conditions. Z is not dominated
by any mixed strategy
→ 𝑍 ∈ 𝐵1
- Therefore, for player 1: 𝑩𝟏 = 𝑼𝑫𝟏 = {𝑿, 𝒀, 𝒁}
Dominant Strategy Equilibrium
- If every player has a dominant strategy, the game has a dominant strategy equilibrium
(solution)
- Dominant Strategy Axiom: if a player has a dominant strategy, he will use it
- Problem with dominant strategy equilibrium: in many games there does not exist one
- Dominant Strategy Equilibrium (DSE): Prisoners’ Dilemma:
- (Confess, Confess) is a dominant strategy equilibrium
Efficiency and Equilibrium
- Game equilibrium is a characterization of the outcome of individually rational behavior
- Because of strategic interaction, rational behaviour does not always lead to outcomes
that are mutually the best
- Dominant strategy equilibrium in Prisoners’ Dilemma: (Confess, Confess)
- But this is not socially efficient: both players are better off with (Not Confess, Not Confess)
Pareto Optimality
- A solution is Pareto optimal if and only if there is no other solution that is:
1. Better for at least one agent
2. No worse for everyone else
- A mild (weak) criterion for social efficiency
- The Prisoner’s Dilemma solution is not Pareto optimal
- Example: (Low, Low) is DSE
What to do When Equilibrium is Inefficient?
- Can’t always be improved (ie. arms race)
- Opportunities:
- Collude/cooperate (could be illegal) (ie. marriage, OPEC); might involve side payments
if not win-win
- Design systems to increase trust
- Repeated interactions: build trust or create punishment scheme
1 1 1
- Practice Question: Suppose player 2 has the belief (2 , 4 , 4) regarding the strategy that player
1 employs. With this belief, find player 2’s best responses
𝟏 𝟏 𝟏
- Answer: 𝑩𝑹𝟐 (𝟐 , 𝟒 , 𝟒) = {𝑳, 𝑹}
- For player 2:
1
1
1
1
1
1
1
1
1
- Expected payoff of L = (2 ∗ 6) + (4 ∗ 3) + (4 ∗ 1) = 4
- Expected payoff of C = (2 ∗ 4) + (4 ∗ 0) + (4 ∗ 5) =
13
4
- Expected payoff of R = (2 ∗ 4) + (4 ∗ 5) + (4 ∗ 3) = 4
More Efficient vs. Pareto Efficient
- (D,R) is more efficient than both (M,L) and (U,R). With (D,R), one player is better and no one
is worse off
- In this game, (U,L) and (D,L) are the only efficient strategy profile. There is no strategy profile
that is more efficient than (U,L) and (D,L)
Chapter 7: Rationalizability and Iterated Dominance
Dominance Solvability
- If every player has a dominant strategy, the game has a dominant strategy equilibrium (DSE)
- In some games, there might not be a dominant strategy, but there are dominated strategies
(ie. bad)
- If we can reach a unique strategy vector by iterated elimination of dominated strategies, the
game is said to be dominance solvable
- Example: Playing Mind Games:
- Neither of player 1’ strategies are dominated
- If you are player 1, which strategy should you play?
- Iterative removal of strictly dominated strategies: Step 1:
- If you are player 1, try to put yourself in the shoes of player 2
- For player 2, X is strictly dominated by Y. So you should assign zero probability
to strategy X
- Iterative removal of strictly dominated strategies: Step 2:
- Knowing that player 2 will never play X, player 1’s rational strategy is to play B.
After we remove X from the payoff matrix, A becomes strictly dominated by B in
the reduced game
- Iterative removal of strictly dominated strategies: Step 3:
- Player 2 knows that player 1 will play B. So player 2’s best response is Z. In the
reduced game, Y is dominated by Z.
- With rational players, the only outcome is strategy profile {(𝑩, 𝒁)}
Rationalizable Strategies
- The procedure is called iterative removal of (strictly) dominated strategies (or iterated
dominance). The set of strategies that survive is called the rationalizable strategies
- Steps:
1. Delete all of the dominated strategies for each player
2. Remove any strategies that are dominated in the reduced game
3. Continue this process until no more strategies can be deleted
- Logic of rationalizability depends on:
- Common knowledge of rationality
- Common knowledge of the game
- Common Knowledge:
- Each player knows, each player knows the others know, each player knows the others
know that they all know…
- It is as though the information is publicly announced while the players are together
- Example: Rationalizability/Iterated Dominance:
- Set of rationalizable strategies is {(𝑀, 𝑍)}
- Steps:
1.
2.
3.
4.
Player 2: X is strictly dominated by a mixed strategy (0, ½, ½). We remove X
In the reduced form, U is dominated by D for player 1. We remove U
In the further reduced form, Y is dominated by Z for player 2. We remove Y
Once knowing that player 2 will choose Z, player 1’s strategy D is dominated
by M. We remove D
- So the set of rationalizable strategies is {(𝑀, 𝑍)}
- Notes:
- In two -player games, dominance and best response imply the same restrictions
- Iterated dominance is identical with the procedure in which strategies that are never
best responses are removed at each round
Strategic Uncertainty
- Rationalizability requires players’ beliefs and behavior be consistent with common knowledge
of rationality
- It does not require that their beliefs be correct
- It does not help solve the strategic uncertainty in coordination games
- Strategic Uncertainty: Battle of the Sexes
- Coordination Game: want to go to an event together, with slightly different
preferences
- Any dominant strategies? NO
- Any dominated strategies? NO
- Example: Stag Hunt
- Any dominant strategies? NO
- Any dominated strategies? NO
- Pareto optimal outcomes? (Stag, Stag)
- Facilitate Coordination
- Focal Point: e.g. lost and found
- Institutions, Rules, Norms: e.g. everyone drives on the right, etc.
- Communication
- Question: Find the Set of Rationalizable Strategies for this Game
- For player 1, F is dominated by a mixture of B and C. Remove F
- In the reduced form, F is dominated by C for player 2. Remove F
- Set of rationalizable strategy profiles is 𝑹 = {𝑪, 𝑩} × {𝑪, 𝑩}
Chapter 8: Location, Partnership, and Social Unrest
Location Game
- Pat (𝑃) and Chris (𝐶) plan to sell cold drinks on a beach
- Assume sunbathers are spread evenly across the beach and will walk to the closest vendor.
The beach is divided into nine regions of equal size
- If Pat locates in region 3 and Chris locates in region 8, then Pat serves to customers in
regions 1 through 5 and Chris serves all those in regions 6 through 9
- Find where C and P would like to locate:
- Player 𝑖’s strategy space 𝑺𝒊 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗} 𝑖 = 𝐶, 𝑃
- To compute the set of rationalizable strategies, we need to perform iterated
dominance:
- If both C and P locate in 1, then they split all of the regions
- If C plays 2 when P chooses 1, then C captures regions 2 through 9
- So, the end regions are dominated by the adjacent ones
𝑢𝑖 (1, 𝑠𝑗 ) < 𝑢𝑖 (2, 𝑠𝑗 ). Thus, 𝑅𝑖1 = {2,3,4,5,6,7,8}
- The set of rationalizable strategy profiles is 𝑹 = {(𝟓, 𝟓)}
- Both locate at the center of the beach
Chapters 9-11: Nash Equilibrium
Chapter 9: Nash Equilibrium
Pure Strategy Nash Equilibrium
- A set of strategies forms a Nash equilibrium if the strategies are best replies to each other
- Recall: a strategy is a best reply to a particular strategy of another player if it gives the
highest payoff against that particular strategy
- A Nash equilibrium is a solution concept which predicts what people will do in a game
- A Nash equilibrium is a strategy profile such that no player can change his strategy to
increase his payoff (given all other players’ strategies)
- Definition: a strategy profile 𝑠 ∈ 𝑆 is a Nash equilibrium if and only if 𝑠𝑖 ∈ 𝐵𝑅𝑖 (𝑠−𝑖 ) for each
player 𝑖. That is, 𝑢𝑖 (𝑠𝑖 , 𝑠−𝑖 ) ≥ 𝑢𝑖 (𝑠𝑖′ , 𝑠−𝑖 ) for every 𝑠𝑖′ ∈ 𝑆𝑖 and each player 𝑖
Steps: How to Find the Pure Strategy Nash Equilibrium
1. Find best response of player 1 to each strategy of player 2, mark 1’s payoff, row by row
2. Find best response of player 2 to each strategy of player 1, mark 2’s payoff, column by
column
3. Any cell with two marks indicates a Nash Equilibrium
- There are two Nash equilibria in the Hawk-Dove game:
(Hawk, Dove) and (Dove, Hawk)
- Notes:
- Nash equilibrium may not be unique
- Nash equilibria are strategy profiles, not payoffs
- ie. if you report the Nash equilibria, you should write (Hawk, Dove) and (Dove,
Hawk), not (1, 4) and (4, 1)
- Practice Example: Matching Pennies:
- There is no Nash equilibrium (NE) in pure strategies
- Practice Example: Prisoners’ Dilemma:
- There is only one Nash equilibrium (not pareto efficient):
(Confess, Confess)
- Use either best response or iterated removal of dominated strategies to generate the
same Nash equilibrium
- Practice Example: Battle of the Sexes:
- 2 Nash equilibria:
(Opera, Opera) and (Movie, Movie)
Strict Nash Equilibrium
- A more stringent version of the equilibrium concept is called Strict Nash equilibrium
- Definition: a strategy profile 𝑠 is called a Strict Nash equilibrium if and only if {𝑠𝑖 } = 𝐵𝑅𝑖 (𝑠−𝑖 )
for each player 𝑖. In words, player 𝑖’s strategy 𝑠𝑖 is the only best response to 𝑠−𝑖
- Practice Example:
- There are 2 Nash equilibria:
(D, P) and (P, D)
- (D,P) is a strict Nash equilibrium since D and P are the only best responses to one
another
- (P,D) is not a strict Nash equilibrium since player D can choose P or D, if player S’s
strategy is D
- Practice Example: Coordination:
- There are 2 strict Nash equilibria:
(A,A) and (B,B)
- Practice Example: Pareto Coordination:
- There are 2 strict Nash equilibria:
(A,A) and (B,B)
- Practice Example: Finding Nash Equilibrium:
- There are 2 Nash equilibria:
(D,H) and (H,D)
- There is only 1 strict Nash equilibrium:
(D,H)
Chapter 10: Oligopoly, Tariffs, Crime, and Voting
Duopoly Games
A Duopoly Game by Calculus
- Two firms set, simultaneously and independently, their prices, 𝒑𝟏 and 𝒑𝟐
- Consumers demand 𝟏𝟎 − 𝒑𝟏 + 𝒑𝟐 units of firm 1’s good and 𝟏𝟎 − 𝒑𝟐 + 𝒑𝟏 units of firm 2’s
good
- Each firm produces at zero cost. The firm’s payoffs are their profits
a) Write the payoff functions of the firms (hint: as a function of their strategies 𝒑𝟏 and 𝒑𝟐 )
b) Compute firm 2’s best-response function (hint: as a function of 𝒑𝟏 ) and firm 1’s bestresponse function (hint: as a function of 𝒑𝟐 ). Illustrate in a graph.
c) Find the Nash equilibrium
- Solution:
a) Production is costless (ie. cost = 0), so firm 𝑖’s profit (or payoff) is:
𝑢1 (𝑝1 , 𝑝2 ) = (10 − 𝑝1 + 𝑝2 ) ∙ 𝑝1 = 10𝑝1 − 𝑝12 + 𝑝2 𝑝1
𝑢2 (𝑝1 , 𝑝2 ) = (10 − 𝑝2 + 𝑝1 ) ∙ 𝑝2 = 10𝑝2 − 𝑝22 + 𝑝1 𝑝2
b) Find the best response of one to the other’s price: we need to find 𝑝1 that
maximizes player 1’s payoff given 𝑝2 . Take the first order condition:
𝜕𝑢1
𝑝2
= 10 − 2𝑝1 + 𝑝2 = 0 ⟶ 𝐵𝑅1 = 𝑝1∗ = 5 +
𝜕𝑝1
2
𝜕𝑢2
𝑝1
= 10 − 2𝑝2 + 𝑝1 = 0 ⟶ 𝐵𝑅2 = 𝑝2∗ = 5 +
𝜕𝑝2
2
c) Solve the linear system (2 equations and 2 unknown variables). Sub 𝑝2∗ = 5 +
𝑝1∗ :
𝑝2∗
∗
𝑝1 = 5 +
2
𝑝∗
(5 + 1 )
2
𝑝1∗ = 5 +
2 ∗
𝑝1
2𝑝1∗ − 10 = 5 +
2
4𝑝1∗ − 30 = 𝑝1∗
3𝑝1∗ = 30
𝑝1∗ = 10
𝑝1
2
into
and 𝑝2∗ = 10
- This is a Nash equilibrium by definition – the only one
Cournot Duopoly Game
- Two firms set, simultaneously and independently, their output level, 𝒒𝟏 and 𝒒𝟐 (measured in
thousands)
- Consumers demand 𝑷 = 𝟏𝟎𝟎𝟎 − 𝑸, where 𝑸 = 𝒒𝟏 + 𝒒𝟐
- Each firm must pay a production cost of $100 per thousand units
a) Write the payoff functions of the firms
b) Compute each firm’s best-response function
c) Find the Nash equilibrium
- Solution:
a) Firm 𝑖’s profit (or payoff) is:
𝑢1 (𝑞1 , 𝑞2 ) = 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑣𝑒𝑛𝑢𝑒1 − 𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡1
𝑢1 (𝑞1 , 𝑞2 ) = 𝑃𝑟𝑖𝑐𝑒 ∙ 𝑞1 − 𝑇𝐶1
𝑢1 (𝑞1 , 𝑞2 ) = (1000 − 𝑞1 − 𝑞2 )𝑞1 − 100𝑞1 = 1000𝑞1 − 𝑞12 − 𝑞2 𝑞1 − 100𝑞1
and
𝑢2 (𝑞1 , 𝑞2 ) = (1000 − 𝑞1 − 𝑞2 )𝑞2 − 100𝑞2 = 1000𝑞2 − 𝑞22 − 𝑞1 𝑞2 − 100𝑞2
b) Find the best response of one to other’s output: we need to find 𝑞1 that maximizes
player 1’s payoff given 𝑞2 . Take the first order condition:
𝜕𝑢1
𝑞2
= 1000 − 2𝑞1 − 𝑞2 − 100 = 0 ⟶ 𝐵𝑅1 = 𝑞1∗ = 450 −
𝜕𝑞1
2
𝜕𝑢2
𝑞1
= 1000 − 2𝑞2 − 𝑞1 − 100 = 0 ⟶ 𝐵𝑅2 = 𝑞2∗ = 450 −
𝜕𝑞2
2
c) Solve the linear system (2 equations and 2 unknown variables) by subbing 𝑞2∗ =
𝑞
450 − 21 into 𝑞1∗ :
𝑞2
𝑞1∗ = 450 −
2
𝑞∗
(450 − 1 )
2
𝑞1∗ = 450 −
2
𝑞1∗
2𝑞1∗ − 900 = −450 +
2
4𝑞1∗ − 900 = 𝑞1∗
3𝑞1∗ = 900
𝑞1∗ = 300
and 𝑞2∗ = 300
- This is a Nash equilibrium by definition – the only one
Bertrand Duopoly Game
- Two firms set, simultaneously and independently, their prices
- The firm with the lowest prices serves the whole market. If both firms offer the same lowest
price, the market is shared. 𝑷 = 𝟏𝟎𝟎𝟎 − 𝑸
- Price can be any non-negative real number. The quantity demanded is a linear function of the
price
- The firm’s payoffs are their profits, which are the difference between revenue and costs pf
production (which is 𝒄 = 𝟏𝟎𝟎 per unit produced)
- Find the Nash equilibrium
- Solution:
- The normal form of the game:
- Players: Firms 1 and 2
- Strategy Sets: 𝑆1 = [0, ∞) and 𝑆2 = [0, ∞)
- Firm 𝑖’s Payoff 𝑢𝑖 (𝑝1 , 𝑝2 ):
𝑃𝑖 < 𝑃𝑗 ⟶
{ 𝑃𝑖 > 𝑃𝑗 ⟶
𝑞𝑖 𝑝𝑖 − 100𝑞𝑖 = (1000 − 𝑝𝑖 )𝑝𝑖 − 100(1000 − 𝑝𝑖 ) = (1000 − 𝑝𝑖 )(𝑝𝑖 − 100)
𝟎
}
𝑃𝑖 = 𝑃𝑗 ⟶
𝑞 𝑖 𝑝𝑖
2
−
100𝑞𝑖
2
=
(1000−𝑝𝑖 )𝑝𝑖
2
−
100(1000−𝑝𝑖 )
2
=
(1000−𝑝𝑖 )(𝑝𝑖 −100)
2
- In equilibrium:
- Neither firm can be pricing below 100 (negative profits)
- 𝑷𝒊 > 𝑷𝒋 ≥ 𝟏𝟎𝟎 cannot be equilibrium (has an incentive to undercut the other)
- 𝑷𝒊 = 𝑷𝒋 > 𝟏𝟎𝟎 cannot be equilibrium (has an incentive to undercut the other)
- The only possible equilibrium prices are: 𝑷𝒊 = 𝑷𝒋 = 𝟏𝟎𝟎
- Each prices at marginal cost in equilibrium
Chapter 11: Mixed Strategy Nash Equilibrium
The Game of Tennis: Serving
- Server chooses to serve either left or right
- Receiver defends either left or right
- Better chance to get a good return if you defend in the area the server is serving to
- Game Table:
- For Server:
- Best response to defend left is to serve right
- Best response to defend right is to serve left
- For Receiver: - Just the opposite
- Nash Equilibrium:
- Notice that there are no mutual best responses in this game
- This means there are no Nash equilibria in pure strategies
- But games like this always have at least one Nash equilibrium
- What are we missing?
- Extended Game:
- Suppose we allow each player to choose randomizing strategies
- For example, the server might serve left half the time and right half the time
- In general, suppose the server serves left a fraction 𝒑 of the time
- What is the receiver’s best response?
- Calculating Best Responses:
- Clearly if 𝑝 = 1, then the receiver should defend to the left
- If 𝑝 = 0, then the receiver should defend to the right
- The expected payoff to the receiver is:
3
1
1
3
(𝑝) ( ) + (1 − 𝑝) ( ) if defending left
4
4
(𝑝) ( ) + (1 − 𝑝) ( ) if defending right
4
4
- Therefore, she should defend left if:
3
1
1
3
(𝑝) ( ) + (1 − 𝑝) ( ) > (𝑝) ( ) + (1 − 𝑝) ( )
4
4
4
4
- Rewriting:
𝑝 >1−𝑝
⟶
𝟏
𝒑>𝟐
- Plot the Receiver’s Best Response:
- Server’s Best Response:
- Suppose that the receiver goes left with probability 𝒒
- Clearly, if 𝑞 = 1, the server should serve right
- If 𝑞 = 0, the server should serve left
- More generally, serve left if:
1
3
3
1
(4) (𝑞) + (4) (1 − 𝑞) > (4) (𝑞) + (4) (1 − 𝑞)
- Simplifying, he should serve left if:
𝟏
𝒒<𝟐
- Plot the Server’s Best Response:
- Putting Best Responses Together:
- Equilibrium:
𝟏 𝟏
- Mixed Strategy Nash Equilibrium: 𝝈𝟏 = (𝟐 , 𝟐) = 𝝈𝟐
Mixed Strategy Equilibrium
- A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses
- In the tennis example, this occurred when each player chose a 50-50 mixture of left and right
- Consider a strategy profile 𝜎 = (𝜎1 , 𝜎2 , … , 𝜎𝑛 ), where 𝜎𝑖 ∈ ∆𝑆𝑖 for each player 𝑖. Profile 𝜎 is a
mixed-strategy Nash equilibrium if and only if 𝑢𝑖 (𝜎𝑖 , 𝜎−𝑖 ) ≥ 𝑢𝑖 (𝑠𝑖′ , 𝜎−𝑖 ) for each 𝑠𝑖′ ∈ 𝑆𝑖 and
each player 𝑖. That is, 𝜎𝑖 is a best response to 𝜎−𝑖 for every player 𝑖
- Summary: Procedure for Finding Mixed Strategy Equilibrium:
1. Calculate the set of rationalizable strategies by performing the iterated-dominance
procedure
2. Restricting attention to rationalizable strategies, write equations for each player to
characterize mixing probabilities that make the other player indifferent between the
relevant pure strategies
3. Solve these equations to determine equilibrium mixing probabilities
- General Properties of Mixed Strategy Equilibria:
- A player chooses his strategy so as to make his rival indifferent
- A player earns the same expected payoff for each pure strategy chosen with positive
probability
- Every finite game (having a finite number of players and a finite strategy space) has at
least one Nash equilibrium in pure or mixed strategies
- Practice Example: Matching Pennies:
- Each player chooses heads or tails and then reveals their coins at the same time. If the
coins match, Player A wins and receives a dollar from Player B. Otherwise, Player B wins
and receives a dollar from Player A
- There is no Nash equilibrium in pure strategies for this game
- No combination of heads or tails leaves both players satisfied – one player or the other
will always want to change strategies
- Let 𝑝 be the probability that A chooses Heads and 𝑞 be the probability that B chooses
Heads
- In this game, nothing is dominated; everything is rationalizable
- A’s Expected Payoff:
- When choosing Heads = (1)(𝑞) + (−1)(1 − 𝑞)
- When choosing Tails = (−1)(𝑞) + (1)(1 − 𝑞)
(1)(𝑞) + (−1)(1 − 𝑞) = (−1)(𝑞) + (1)(1 − 𝑞)
𝑞 + 𝑞 − 1 = −𝑞 − 𝑞 + 1
2𝑞 − 1 = −2𝑞 + 1
4𝑞 = 2
𝟏
𝒒∗ = 𝟐
- B’s Expected Payoff:
- When choosing Heads = (−1)(𝑝) + (1)(1 − 𝑝)
- When choosing Tails = (1)(𝑝) + (−1)(1 − 𝑝)
(−1)(𝑝) + (1)(1 − 𝑝) = (1)(𝑝) + (−1)(1 − 𝑝)
−𝑝 − 𝑝 + 1 = 𝑝 + 𝑝 − 1
−2𝑝 + 1 = 2𝑝 − 1
4𝑝 = 2
𝟏
𝒑∗ = 𝟐
- Mixed Strategy Nash Equilibrium:
𝟏 𝟏
𝝈𝟏 = (𝟐 , 𝟐) = 𝝈𝟐
- Practice Example: Find the Mixed-Strategy Nash Equilibrium:
- Solution:
1. For 1, F dominated by 𝜎1 = (0, 0.25, 0.75). Deleting F, turn to player 2: F is
dominated by C. No further deletions, 𝑅 = {𝐶, 𝐵} × {𝐶, 𝐵}
2. 𝜎1 = (0, 𝑝, 1 − 𝑝), 𝜎2 = (0, 𝑞, 1 − 𝑞)
3. Seek 𝑝, 𝑞 that make players indifferent:
(𝑞)(0) + (1 − 𝑞)(3) = 𝑢1 (𝐶, 𝜎2 ) = 𝑢1 (𝐵, 𝜎2 ) = (𝑞)(3) + (1 − 𝑞)(2)
𝟏
𝒒=
𝟒
(5)(𝑝) + (2)(1 − 𝑝) = 𝑢2 (𝐶, 𝜎1 ) = 𝜇2 (𝐵, 𝜎1 ) = (2)(𝑝) + (3)(1 − 𝑝)
𝒑 = 𝟏/𝟒
1 3
- Conclusion: Game has a sole Nash equilibrium, 𝜎1 = (0, , ) = 𝜎2
4 4
Chapter 12: Strictly Competitive Games & Security Strategies
Chapter 12: Strictly Competitive Games and Security Strategies
Strictly Competitive Game
- A two-player, strictly competitive game is a two-player game with the property that for every
two strategy profiles 𝑠, 𝑠 ′ ∈ 𝑆, 𝑢1 (𝑠) > 𝑢1 (𝑠 ′ ) if and only if 𝑢2 (𝑠) < 𝑢2 (𝑠 ′ )
- ie. player 1’s payoff increases, if and only if, player 2’s payoff decreases
- In a strictly competitive game, the two players have exactly opposite rankings over the
outcomes. Wherever one player’s payoff increases, the other one’s payoff decreases
- Such games offer no room for joint gain or compromise
- Examples: matching pennies, chess, checkers, tennis, football
- Please note: the matching pennies game is a special type of strictly competitive game
called zero-sum, in which the players payoffs always sum to zero
- Strictly Competitive Game: Matching Pennies
- Zero-Sum Game: sum of payoffs in each cell is zero
- Example: A Two-Player, Strictly Competitive Game
- Practice: Are the following games strictly competitive?
Security Strategy
- The concept of a security strategy is based on evaluating “worst-case scenarios” by focusing
on each player’s own strategies. In any game, the worst payoff that player 𝑖 can get when
playing strategy 𝑠𝑖 is defined by:
𝒘𝒊 (𝒔𝒊 ) ≡ 𝐦𝐢𝐧𝐬𝐣∈𝑺𝒋 𝒖𝒊 (𝒔𝒊 , 𝒔𝒋 )
- A strategy 𝑠𝑖 ∈ 𝑆𝑖 for player 𝑖 is called a security strategy if 𝑠𝑖 solves max 𝑤𝑖 (𝑠𝑖 )
𝑠𝑖 ∈𝑆𝑖
- Player 𝑖’s security payoff level is max 𝑤𝑖 (𝑠𝑖 )
𝑠𝑖 ∈𝑆𝑖
- A security strategy gives a player the best of the worst cases
- Payoff level can also be written as:
max min 𝑢𝑖 (𝑠𝑖 , 𝑠𝑗 )
𝑠𝑖 ∈𝑆𝑖 𝑠𝑗 ∈𝑆𝑗
- Example:
- Player 1:
- Playing A: the lowest payoff is 0
- Playing B: the lowest payoff is 1
- The greater of these two payoffs is 1
- Player 1’s security strategy is B and his security level is 1
- Player 2:
- Playing X: the lowest payoff is 1
- Playing Y: the lowest payoff is 3
- The greater of these two payoffs is 3
- Player 2’s security strategy is Y and his security level is 3
Maxmin Strategy
- Note that the definition of a security strategy is formulated in terms of a pure strategy for
player 𝑖
- Another version of the concept focuses on mixed strategies. To differentiate it from the purestrategy version, it is called the “maxmin strategy” in this textbook
- A mixed strategy 𝜎𝑖 ∈ ∆𝑆𝑖 for player 𝑖 is called a maxmin strategy if 𝜎𝑖 solves
max min 𝑢𝑖 (𝜎𝑖 , 𝑠𝑗 ). Player 𝑖’s maxmin payoff level is the value max min 𝑢𝑖 (𝜎𝑖 , 𝑠𝑗 )
σi ∈∆𝑆𝑖 𝑠𝑗 ∈𝑆𝑗
σi ∈∆𝑆𝑖 𝑠𝑗 ∈𝑆𝑗
- Notes:
- The security strategy identifies a payoff that can be certain to achieve at minimum,
whereas maxmin strategy identifies a lower bound that one can achieve in expectation
(or, on average)
- If 𝑠𝑖 is a Nash Equilibrium strategy in a strictly competitive game, then 𝑠𝑖 guarantees
player 𝑖 at least his security payoff level, and player 𝑖 cannot improve this lower bound
by randomizing
- Unlike the Nash Equilibrium, the security/maxmin solution does not require players to
react to a rival’s choice. Such strategy is conservative and usually is not profit
maximizing, but it can be a good choice if a player thinks his rival may not behave
rationally
- This solution is more likely than the Nash solution in cases where there is a
higher probability of irrational (non-optimizing) behavior
- Security strategies may not be rationalizable
Chapters 14-17: Backward Induction and Subgame Perfection
Nash Equilibrium (SPNE)
Chapter 14: Details of the Extensive Form
Game Trees: Basic Language
- Recall:
- Trees consist of nodes connected by branches (arrows)
- Nodes may have successors; if so, have immediate successors
- Nodes may have predecessors; if so, have immediate predecessors
- Clearly, one node is a successor of another node if and only if the other is a
predecessor of this node
- ie. if node a is a predecessor of node b, and node b is a predecessor of node c,
then node a is a predecessor of node c
- An extensive form game is a tree satisfying rules 1-5:
1. Every node is a successor of the initial node, and the initial node is the only one with this
property
2. Each node except the initial node has exactly one immediate predecessor
3. Different branches from a common node must be labeled differently
4. Each information set contains decision nodes for only one of the players
5. All nodes in a given information set must have the same number of immediate
successors and they must have the same set of action labels on the branches leading to
these successors
- Game tree (a) violates Tree Rule 4:
- Game tree (b) violates Tree Rule 5:
Recall vs. Information
- Please note, we generally assume perfect recall: every player, at every information set, can
remember their own past actions and can recall whatever she has observed
- ie. the following tree depicts imperfect recall: player 1 cannot recall, when choosing
between X and Y, whether she has played U or D
- There is perfect information if every information set is a singleton (ie. one-node set, no
dashed lines in the picture)
- There is imperfect information if some information set has multiple nodes
- There is at least one contingency where the player on the move does not know exactly
where he is in the tree
- Perfect recall coexists with perfect/imperfect information
Drawing Infinite Decisions
- Sometimes, a player may have an infinite number of actions. We draw an arc to represent
them in such a case
- ie. Firm 1 decides how much to spend on advertising (between 0 and 1, measured in
million dollars), and then firm 2, after observing 1’s choice, decides whether to exit or
stay in the market
- Let a be firm 1’s action. The payoffs of the players depend on a
- In the following case, we assume that player 2 does not observe the advertising level of
firm 1
Ultimatum-Offer Bargaining
- The seller (player 1) chooses an ultimatum (take-it-or-leave-it) offer; any price $0 to $100
- The seller’s strategy is: 𝑝 ∈ [0, 100]
- The buyer (payer 2) chooses, in each contingency, between “yes” and “no”
- The buyer’s strategy: a function: 𝑠2 : [0, 100] → {𝑌𝑒𝑠, 𝑁𝑜}
- The buyer has an infinite number of information sets, one for each of the feasible
offers of player 1
- A type of strategy for player 2 is setting a “cutoff rule”
Yes
if 𝑝 ≤ 𝑝_
𝑠2 (𝑝) = {
No
if p > p_
- At a terminal node with “Yes”, payoffs are p to the seller and 100 – p to the buyer (assume the
painting’s value is $100 to the buyer)
- At a terminal node with “No”, payoffs are 0 to both (assume the painting is worth nothing to
the seller)
- There are many Nash equilibria. Fix and 𝑃𝑐𝑢𝑡𝑜𝑓𝑓 ∈ [0, 100]
- (𝑃, 𝑃𝑐𝑢𝑡𝑜𝑓𝑓 ) if a Nash equilibrium with trade
- Player 1: Given player 2’s strategy, can 1 do better?
- By lowering price, seller still sells but at less, so worse off. By raising price, seller gets
rejection and gets nothing, no better. So this is the best response, given player 2’s
strategy
- Check for player 2: Given 1’s strategy, can 2 do better?
- It is getting 100 − 𝑃 ≥ 0. Whatever other strategy, it must specify “yes” or “no” at 1’s
P. If yes, 2 gets the same; if no, it gets 0, no more
Chapter 15: Sequential Rationality and Subgame Perfection
Comparison of Normal and Extensive Forms
- An extensive form can be translated into a normal form
- Normal-form concepts are valid for every game
- Nash equilibrium is still a valid solution concept for an extensive form
- In some games, the extensive form most precisely captures the order of moves and the
information structure
Example: Entry and Predation
- There is an incumbent. Potential entrant chooses to enter or stay out. If in, the incumbent
chooses to accommodate (both get modest profits) or to trigger a price war (both suffer)
- Player 1: potential entrant
- Player 2: incumbent
- Write the Normal Form for the Entry and Predation Game:
- Two pure strategy Nash equilibria: (I, A) and (O, P)
- However, (O, P) is not sequentially rational:
- If player 1 mistakenly enters, then, it is best for player 2 to accommodate (+2)
rather than start a price war (-1)
Rationality: Ex Ante vs. Sequential
- Nash equilibrium is a fine concept form normal-form games where players move
independently
- When a normal-form game is an extensive form game translation with dependent moves,
Nash equilibrium may fail to capture the sequential rationality
- In the previous game, (O, P) is a NE and is ex ante rational: from the standpoint of the initial
node and strategies
- (O, P) is not sequentially rational: from the standpoint of information set of player 2 (even as
(O, P) does not lead there)
Sequential Rationality
- An optimal strategy for a player should maximize his expected payoff, conditional on every
information set at which this player has the move
- Player 𝑖’s strategy should specify an optimal action from each of player 𝑖’s information sets,
even those that player 𝑖 does not believe (ie. ex ante) will be reached in the game
- In this sense, incumbent’s Price War is not sequentially rational
Backward Induction
- We restrict attention to games of perfect information (singletons)
- If sequential rationality is common knowledge, each player should look ahead to consider
what others will do in response to his move at a particular information set
- Backward induction: the process of analyzing a game from back to front (from information
sets at the end of the tree to those at the beginning). At each information set, strike out actions
that are dominated, given the terminal nodes that can be reached
- Example: Backward Induction
- Arrows indicate optimal action at each information set
- Proper Subgame: player 2 decides between A and B: 4 > 2 therefore player 2 chooses A
- Replace whole proper subgame with (1, 4)
- Proper Subgame: player 1 decides between E and F: 3 > 2 therefore player 1 chooses E
- Replace whole proper subgame with (3, 3)
- Proper Subgame: player 2 decides between C (3, 3) and D (6, 2): 3 > 2, so p2 chooses C
- Replace whole proper subgame with (3, 3)
- Subgame: player 1 decides between U (1, 4) and D (3, 3): 3 > 1, so player 1 chooses D
- A single sequentially rational strategy profile: (DE, AC)
- This is also a Nash equilibrium
- Exercise: Voting
- Three legislators are voting on whether to give themselves a pay raise. All three want
the pay raise; however, each faces a small cost 𝑐 (𝑐 > 0)
- The benefit for the raise is greater than the cost: 𝑏 > 𝑐
- They vote in the order 1-2-3. Simple majority rule. What is the outcome obtained by
backward induction?
- Proper Subgame: 3 chooses between 𝑦 and 𝑛: 𝑏 > 𝑏 − 𝑐: chooses 𝑛
- Replace proper subgame with (𝑏 − 𝑐, 𝑏 − 𝑐, 𝑏)
- Proper Subgame: 3 chooses between 𝑦′ and 𝑛′: 𝑏 − 𝑐 > 0: chooses 𝑦′
- Replace proper subgame with (𝑏 − 𝑐, 𝑏, 𝑏 − 𝑐)
- Proper Subgame: 3 chooses between 𝑦′′ and 𝑛′′: 𝑏 − 𝑐 > 0: chooses 𝑦′′
- Replace proper subgame with (𝑏, 𝑏 − 𝑐, 𝑏 − 𝑐)
- Proper Subgame: 3 chooses between 𝑦′′′ and 𝑛′′′: 0 > −𝑐: chooses 𝑛′′′
- Replace proper subgame with (0, 0, 0)
- Proper Subgame: 2 chooses between 𝑌 (𝑏 − 𝑐, 𝑏 − 𝑐, 𝑏) and 𝑁 (𝑏 − 𝑐, 𝑏, 𝑏 − 𝑐)
𝑏 > 𝑏 − 𝑐: chooses 𝑁
- Replace proper subgame with (𝑏 − 𝑐, 𝑏, 𝑏 − 𝑐)
- Proper Subgame: 2 chooses between 𝑌 ′ (𝑏, 𝑏 − 𝑐, 𝑏 − 𝑐) and 𝑁′(0,0,0)
𝑏 − 𝑐 > 0: chooses 𝑌′
- Replace proper subgame with (𝑏, 𝑏 − 𝑐, 𝑏 − 𝑐)
- Subgame: 1 chooses between 𝑌 (𝑏 − 𝑐, 𝑏, 𝑏 − 𝑐) and 𝑁 (𝑏, 𝑏 − 𝑐, 𝑏 − 𝑐)
𝑏 > 𝑏 − 𝑐: 1 chooses 𝑁
- Therefore, SPNE: (𝑁, 𝑁𝑌 ′ , 𝑛𝑦 ′ 𝑦 ′′ 𝑛′′′ )
Backward Induction and Nash Equilibrium
- Backward induction identifies a unique path of actions, provided there are no terminal nodes
at which some player gets the same payoff
- Every finite extensive-form game of perfect information has a pure-strategy Nash equilibrium
that can be identified by backward induction. If no player has the same payoffs at any two
terminal nodes, then there is a unique Nash equilibrium that can be so identified
- Example: Advertising and Entry
- A retailer is facing possible competition from a potential entrant. He can deter entry by
engaging in an advertising and price cutting campaign
- The rival is fast and flexible, so its policy is to wait and decide at the very last second its
entry choice
- The rival observes the start of this campaign before making its entry decision
- Incumbent: Advertise or Not
- Rival: Enter or Not
- At equilibrium, the incumbent will run its campaign and this will effectively
deter entry
- Suppose that the rival must commit to enter or not before the advertising decision of
the incumbent (note: rival is now player 1, and incumbent is now player 2)
- At equilibrium, the rival will enter and the incumbent will not advertise
- This game has a first-mover advantage
- Note: Not all games exhibit first-mover advantage
- There is second-mover advantage
- Examples: setting price, bidding
- Clearly it pays to go second and beat the bid of the first firm
Potential Problems with Backward Induction
- If a player has the same payoffs at any two terminal nodes, then a unique Nash equilibrium
cannot be identified
Subgame and Proper Subgame
- Given an extensive-form game, a node 𝑥 in the tree is said to initiate a subgame if neither 𝑥
(nor any of its successors) are in an information set that contains nodes that are not successors
of 𝑥
- A subgame is the tree structure defined by such a node 𝑥 and its successors
- Subgames that start from nodes other than the initial node are called proper subgames
- Note: the original game is always a subgame
- In a game of perfect information, every node initiates a subgame
- Practice: How many subgames does the game tree have?
- In a game of perfect information, every node initiates a subgame
- There are 6 proper subgames and 7 subgames
- Example: Game with Imperfect Information
- This game tree has 1 proper subgame (starting at node 𝑦) and 2 subgames (the proper
subgame and the whole game)
Subgame Perfect Nash Equilibrium (SPNE)
- A strategy profile is a SPNE if it specifies a Nash Equilibrium in every subgame of the original
game
- A solution concept should be consistent with its own application from anywhere in the game
where it can be applied
- Key Points:
- A SPNE is a Nash Equilibrium (a refinement of NE)
- For games of perfect information, backward induction yields SPNE
- Backward induction may apply with imperfect information. However, there are cases
where imperfect information could be an obstacle to backward induction
Finding SPNE
- Using backward induction to find the SPNE:
- Start from the subgames which start with a node closest to a terminal node
- Find Nash Equilibrium of the subgame
- Replace the subgame with the Nash Equilibrium payoffs and work backwards
- If there are more than one Nash Equilibria of the subgame, repeat this for each
subgame
- Practice:
- Backward induction leads to a unique SPNE: (OA, O)
- Note: sequential rationality leads to joint stupidity
- Example 1: What are the NE and SPNE of this Game?
- One proper subgame and two subgames
- Two NE: (D, R) and (U, L)
- Consider the proper subgame: the unique NE of the subgame is R.
- Therefore, (U, L) is not a SPNE
- (D, R) is the only SPNE
- Note: SPNE coincides with the outcome obtained from backward induction
- Example 2 (a): Subgame Perfection
- 1 proper subgame and 2 subgames
(c) Normal Form of the Proper Subgame
- One pure strategy NE: (A, X)
(b) Normal Form of the Entire Game
- The entire game has 3 pure strategy NE:
(UA, X), (DA, Y) and (DB, Y)
- The entire game has 3 pure strategy NE: (UA, X), (DA, Y), and (DB, Y)
- Only (UA, X) is the SPNE
- Key Point: The subgame perfection concept requires equilibrium at every subgame,
meaning that if any particular subgame is reached, then we can expect the players to
follow through with the prescription of the strategy
- Example 3: Price War
- NE: (NH, H) and (NL, H)
- How many proper subgames? – 1
- Which of the NE are subgame perfect?
- Unique NE of subgame is (H, H)
- Since (L, H) is not a NE of the subgame, (NL, H) is not a SPNE
- The only SPNE is (NH, H)
- Example 4: Multiple SPNE
- The proper subgame has two NE:
- (u, L) yielding (4, 2)
- (d, R) yielding (1, 2)
- Case 1: Replacing subgame with payoff from (u, L)
- First SPNE: (Du, L) yielding (4, 2)
- Case 2: Replacing subgame with payoff from (d, R)
- Second SPNE: (Ud, R) yielding (2, 1)
- Example 5: Multiple SPNE
- 2 proper subgames:
- First proper subgame:
- 1-person decision problem. NE (A) yields (2, 2, 2)
- Second proper subgame:
- Yields two NE: (W, X) and (Z, Y)
- Case 1:
- Replace the top proper subgame with payoff from playing A (2,2,2)
- Replace the bottom proper subgame with payoff from playing (W,X): (4,2,1)
- SPNE: (UW, L, AX) yields (2,2,2)
(DW, R, AX) yields (4,2,1)
- Case 2:
- Replace the top proper subgame with payoff from playing A (2,2,2)
- Replace the bottom proper subgame with payoff from playing (Z,Y): (1,0,3)
- SPNE: (UZ, L, AY) yields (2,2,2)
Summary of SPNE
- A SPNE is a refinement of NE
- For games of perfect information, backward induction yields SPNE
- The procedure we described is usually the best procedure for finite games
The Stackelberg Duopoly Game
- Two firms move sequentially rather than simultaneously. Firm 1 selects 𝒒𝟏 and this is observe
by firm 2. Then firm 2 selects 𝒒𝟐
- Consumers’ demand 𝒑 = 𝟏𝟐 − 𝒒𝟏 − 𝒒𝟐 . Each firm produces at zero cost. The firms’ payoffs
are their profits
a) Find the SPNE of this game
b) Draw the extensive form of this game
- Solution:
- Production is costless, so firm 𝑖’s profit (or payoff) is:
𝑢1 (𝑞1 , 𝑞2 ) = (12 − 𝑞1 − 𝑞2 )𝑞1 = 12𝑞1 − 𝑞12 − 𝑞2 𝑞1
𝑢2 (𝑞1 , 𝑞2 ) = (12 − 𝑞1 − 𝑞2 )𝑞2 = 12𝑞2 − 𝑞1 𝑞2 − 𝑞22
- We’ll find the SPNE using backward induction. Let’s focus on firm 2 first. We
need to find 𝑞2 that maximizes firm 2’s payoff given 𝑞1
- Take the first order condition:
𝜕𝑢2
𝜕𝑞2
= 12 − 𝑞1 − 2𝑞2 = 0
→ 𝐵𝑅2 = 𝑞2∗ =
12−𝑞1
2
- Keep in mind that the quantity cannot be negative: 𝐵𝑅2 = 𝑞2∗ = max {
12−𝑞1
2
, 0}
- Firm 1 knows firm 2’s best response. Firm 1 chooses 𝑞1 that maximizes:
𝟏𝟐−𝒒𝟏
12−𝑞1
𝑢1 (𝑞1 , 𝑞2∗ ) = (12 − 𝑞1 − 𝑞2∗ )𝑞1 = (12 − 𝑞1 −
) 𝑞1 = 12𝑞1 − 𝑞12 −
𝑞1
𝟐
𝜕𝑢1
𝜕𝑞1
2
= 12 − 2𝑞1 − 6 + 𝑞1 = 0
→ 𝑞1∗ = 6
- Given 𝐵𝑅2 = 𝑞2∗ = max {
12−𝑞1
2
, 0}, then 𝐵𝑅2 = 𝑞2∗ = 3
- SPNE: Firm 1 produces 6 units and firm 2 produces 3 units
- The Extensive Form of This Game:
Chapter 16: Applications of Sequential Rationality and Subgame
Perfection: Topics in Industrial Organization
Advertising Game and Competition
- There are two firms in a market. Firm 1 selects an advertising level 𝒂 (𝒂 ≥ 𝟎) and this is
observed by firm 2. Advertising has a positive effect on the demand for the good sold in the
industry, enhancing the price that the consumers are willing to pay for the output of both firms
- Consumers’ demand 𝒑 = 𝒂 − 𝒒𝟏 − 𝒒𝟐 . After firm 1 selects 𝒂, it is observed by firm 2. Then
the two firms simultaneously and independently select their output levels
- Assume firm 2 produces at zero cost. Firm 1 must pay an advertising cost of 𝟐 𝒂𝟑 /𝟖𝟏. The
firm’s payoffs are their profits.
- Find the SPNE of this game
- Solution:
- Suppose a subgame is reached following advertisement level 𝒂 selected by firm 1. Firm
1’s profit (or payoff) is:
𝒖𝟏 (𝒂, 𝒒𝟏 , 𝒒𝟐 ) = (𝒂 − 𝒒𝟏 − 𝒒𝟐 )𝒒𝟏 − 𝟐 𝒂𝟑 /𝟖𝟏
- Take the first order condition:
𝜕𝑢1
𝑎−𝑞
= 𝑎 − 2𝑞1 − 𝑞2 = 0 → 𝐵𝑅1 = 𝑞1∗ = 2 2
𝜕𝑞
1
- Likewise, 𝐵𝑅2 = 𝑞2∗ =
𝑎−𝑞1
2
- Solving this linear system , we find 𝑞1 = 𝑞2 = 𝑎/3, the equilibrium price 𝑝 = 𝑎/3
- Plugging these values into the firms’ profit functions. Firm 1’s profit (or payoff) is:
𝑢1 (𝑎, 𝑞1 , 𝑞2 ) = (𝑎 − 𝑞1 − 𝑞2 )𝑞1 − 2 𝑎3 /81 = 𝑎2 /9 − 2𝑎3 /81
- Take the first order condition:
𝜕𝑢1
𝜕𝑎
=
2𝑎
9
−
6𝑎2
81
=0
- Solving for 𝑎: 𝑎 ∗ = 3
𝑎
- SPNE: 𝑎∗ = 3, 𝑞1∗ = 𝑞2∗ = 3 = 1
Chapter 17: Applications of Sequential Rationality and Subgame
Perfection: Parlor Games
- No lecture notes, see textbook
Chapters 22-23: Repeated Games
Chapter 22: Repeated Games and Reputation
Dynamic Game
- People often interact in ongoing relationships
- Employment relationships
- Countries competing over tariff levels
- Players condition their decisions on the history of their relationship
- An employee may work diligently only if his employer gave him a good bonus in the
preceding month
- A country may set a low import tariff only if its trading partners maintained low tariffs
in the past
- New Dimension: Time
Reputation
- A person’s past actions affect future beliefs and behavior
- The concern for reputation may motivate parties to cooperate with one another, even
if such behavior requires foregoing short-term gains
- Question: What strategies can lead players to cooperate?
Repeated Games: Definitions
- A repeated game is played over discrete periods of time (period 1, period 2, and so on)
- 𝑡 denotes any given period
- 𝑇 denotes the total number of periods in the repeated game
- 𝑇 can be a finite number, or infinity (ie. played perpetually)
- In each period, players play a static stage game, whereby they simultaneously and
independently select actions
- History of play is observed (ie. the sequence of action profiles)
- The payoff of the entire game is defined as the sum of the stage game payoffs in
periods 1 through 𝑇
A Two-Period Repeated Game (T=2)
- 𝑻 = 𝟐: players 1 and 2 play the above game twice
- Stage Game Nash Equilibrium: (A, Z) and (B, Y)
- The payoff for the entire game is the sum of the stage-game payoffs in the two periods
- Subgame Following (A, Z)
- The subgame following (A, Z) with payoffs (1, 4)
- This matrix is constructed by adding the payoff vector (1, 4) to each of the cells
in the stage game
- Repeated Game Payoffs:
- Altogether, there are 10 possible repeated game payoffs
Stage Nash Profile and SPNE
- ie. Select stage game NE: (A, Z), (B, Y) with payoffs (1,4), (2,1)
- Consider any repeated game: any sequence of stage Nash profiles can be supported as the
outcome of a SPNE
- Every SPNE must specify that in the 2nd period, a stage Nash Equilibrium is selected
- Question: Is there a SPNE that stipulates actions that are not stage Nash Equilibrium in the
first stage?
- ie. Select (A, X) in period 1, then (1) if 2 not deviate from X, select (A, Z), (2) otherwise
play (B, Y) in period 2
- Is it SPNE?
- Select (A, X) in period 1, then (1) if 2 not deviate from X, select (A, Z), (2) otherwise play
(B, Y) in period 2
- 𝑡 = 1: If (A, X) is played, the payoff vector is (4,3)
- Assume player 1 follows but 2 cheats and gets 4 by picking Z
- Player 1 retaliates by choosing B in 𝑡 = 2
- Then 2’s best reaction in 𝑡 = 2 is Y and gets 1
- For player 2, cheating: 4 + 1 = 5 < obeying: get 3 + 4 = 7
- So player 2 has no incentive to cheat
- Similarly, player 1 has no incentive to cheat
- So, YES, the above strategy profile is SPNE
A Two-Period Repeated Game: Reputational Equilibrium as SPNE
- Reputational Equilibrium:
- Non-Stage-Nash profile in 1st period
- Only Stage Nash profile in 2nd period
- 2nd period actions contingent on outcome in first period (whether players cheat or not)
- Example of a reputational equilibrium:
- Select (A, X) in 1st period
- If player 2 chooses X in 1st period, select (A, Z) in 2nd period
- If player 2 chooses Y or Z in 1st period, select (B, Y) in 2nd period
Infinitely Repeated Game (𝑻 = ∞)
- The stage game is played each period for an infinite number of periods
- Discounting Factor (𝜹): future payoffs are not as valuable as current payoffs (𝛿 is a number
between 0 and 1)
- Interest Rate: 𝒓
𝟏
- Discounting Factor: 𝜹 =
𝟏+𝒓
- Review: Discounting:
- Discounting: present-day value of future profits is less than value of current profits
- 𝑟 is the interest rate
- Invest $1 today → get $(1 + 𝑟) next year
$1
- Need $ 1 next year → invest 1+𝑟 today
1
- Note: 1+𝑟 (1 + 𝑟) = 1
- Review: Infinite Sums:
1
1 + 𝛿 + 𝛿 2 + 𝛿 3 + 𝛿 4 + ⋯ = 1−𝛿
1
𝑎
𝑎 + 𝑎𝛿 + 𝑎𝛿 2 + 𝑎𝛿 3 + ⋯ = 𝑎(1 + 𝛿 + 𝛿 2 + 𝛿 3 + ⋯ ) = 𝑎 (1−𝛿) = 1−𝛿
- Exercise:
1
1
1
1
1 + 1+𝑟 + (1+𝑟)2 + (1+𝑟)3 + ⋯ = 1 + 𝑟
1
- In this case, 𝛿 = 1+𝑟
1
- Therefore, 1−𝛿 =
1
1
1−(
)
1+𝑟
1
= 1+𝑟
Prisoner’s Dilemma
- Nash Equilibrium: (Low, Low)
- Cooperation would generate higher payoffs for both players
- Private Rationality (Collective Irrationality):
- The equilibrium that arises from using dominant strategies is worse for every player
than the outcome that would arise if every player used his dominated strategy instead
- Goal:
- To sustain mutually beneficial cooperative outcome that overcomes incentives to
cheat
- Why does the dilemma occur?
- Interaction:
- No fear of punishment
- Exploit repeated play
- Short term or myopic play
- Introduce repeated encounters
- Introduce uncertainty
Long-Term Interaction
- No last period → no backward induction
- Use history-dependent strategies
- Trigger Strategies:
- Begin by cooperating
- Cooperate as long as the rivals do
- Upon observing a defection, immediately revert to a period of punishment of specified
length in which everyone plays non-cooperatively
Two Trigger Strategies
- Grim Trigger Strategy:
- Cooperate until a rival deviates
- Once a deviation occurs, play non-cooperatively for the rest of the game
- Tit-for-Tat:
- Cooperate if your rival cooperated in the most recent period
- Cheat if your rival cheated in the most recent period
- Trigger Strategy Extremes:
- Tit-for-Tat
- Most forgiving
- Shortest memory
- Credible, but lacks deterrence
- Tit-for-tat answers:
“is cooperation easy?”
- Grim Trigger:
- Least forgiving
- Longest memory
- Adequate deterrence but lacks
credibility
- Grim trigger answers:
“is cooperation possible?”
- Why Cooperate (Grim Trigger Strategy)?
- Cooperate if the present value of cooperation is greater than the present value of
defection
- Cooperate: obtain 60 today, 60 next year, 60 the following year, …
- Defect:
obtain 72 today, 54 next year, 54 the following year, …
- Payoff Stream: Grim Trigger Strategy
- Grim Trigger Strategy
- Cooperate If:
𝑷𝑽(𝒄𝒐𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏)
60 + 60𝛿 + 60𝛿 2 + 60𝛿 3 + ⋯
60/(1 − 𝛿)
60
18𝛿
𝛿
>
>
>
>
>
>
𝑷𝑽(𝒅𝒆𝒇𝒆𝒄𝒕𝒊𝒐𝒏)
72 + 54𝛿 + 54𝛿 2 + 54𝛿 3 + ⋯
72 + 54𝛿/(1 − 𝛿)
72(1 − 𝛿) + 54𝛿
12
2/3
- Cooperation is sustainable using grim trigger strategies as long as 𝛿 > 2/3
- The Grim Trigger Strategy is SPNE if 𝜹 > 𝟐/𝟑
- The infinitely repeated game demonstrates that patience (ie. valuing the
future) is essential to an effective reputation
What if a Player Cheats in 𝒕 = 𝟑?
- Cooperate if:
>
𝑃𝑉(𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛)
𝑃𝑉(𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑜𝑛)
2
60 + 60𝛿 + 60𝛿
60 + 60𝛿 + 60𝛿 2
>
3
4
5
+60𝛿 + 60𝛿 + 60𝛿 + ⋯
+72𝛿 3 + 54𝛿 4 + 54𝛿 5 + ⋯
Subtracting (60 + 60𝛿 + 60𝛿 2 ) from both sides
>
60𝛿 3 + 60𝛿 4 + 60𝛿 5 + 60𝛿 6 …
72𝛿 3 + 54𝛿 4 + 54𝛿 5 + 54𝛿 6 …
Divide both sides by 𝛿 3
>
60 + 60𝛿 + 60𝛿 2 + 60𝛿 3 + ⋯
72 + 54𝛿 + 54𝛿 2 + 54𝛿 3 + ⋯
>
60/(1 − 𝛿)
72 + 54𝛿/(1 − 𝛿)
60
>
72(1 − 𝛿) + 54𝛿
>
12
18𝛿
>
𝛿
2/3
- Cooperation is sustainable using grim trigger strategies as long as 𝛿 > 2/3
- The Grim Triger Strategy is SPNE if 𝜹 > 𝟐/𝟑
Tit-for-Tat
- Cooperate if:
𝑃𝑉(𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛)
𝑃𝑉(𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛)
60 + 60𝛿 + 60𝛿 2 + 60𝛿 3 + ⋯
60 + 60𝛿
13𝛿
𝛿
>
and
>
>
>
>
>
𝑃𝑉(𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑜𝑛)
𝑃𝑉(𝑑𝑒𝑓𝑒𝑐𝑡 𝑜𝑛𝑐𝑒)
72 + 47𝛿 + 60𝛿 2 + 60𝛿 3 + ⋯
72 + 47𝛿
12
12/13
- Much harder to sustain than grim trigger; cooperation may not be likely
- Payoff Stream: Tit-for-Tat
Trigger Strategies
- Grim Trigger and Tit-for-Tat are extremes
- Balance two goals:
- Deterrence:
- GTS is adequate punishment
- Tit-for-tat might be too little
- Credibility:
- GTS hurts the punisher too much
- Tit-for-tat is credible
- Another Example
- The only stage Nash Equilibrium: (D, D)
- When Cooperation can be Sustained: Grim Trigger
- Let’s check whether Grimm Trigger (GT) can form a SPNE:
- Suppose 𝑗 plays GT. If 𝑖 also plays GT, his payoff is:
4
4 + 4𝛿 + 4𝛿 2 + 4𝛿 3 + ⋯ = 1−𝛿
- If 𝑖 defects, he gets 6 in the period of defection, and 0 afterwards. Player 𝑖 has
an incentive to cooperate if:
4
≥6
1−𝛿
𝛿 ≥ 1/3
Modified Grim Trigger (MGT)
- Players alternate between (C, C) and (D, C) over time, starting with (C, C)
- If either or both deviates from the alternating strategy, both will revert to the stage Nash
profile, (D, D)
- Question: Can MGT be supported as a SPNE?
- Suppose 2 plays MGT. If 1 also plays MGT, 1’s payoff is:
𝑃𝑉1 = 4 + 6𝛿 + 4𝛿 2 + 6𝛿 3 + 4𝛿 4 + 6𝛿 5 + 4𝛿 6 + 6𝛿 7 …
= 4(1 + 𝛿 2 + 𝛿 4 + 𝛿 6 + ⋯ ) + 6𝛿(1 + 𝛿 2 + 𝛿 4 + 𝛿 6 + ⋯ )
1
1
= 4 (1−𝛿2) + 6𝛿 (1−𝛿2)
=
4+6𝛿
1−𝛿 2
- If 2 plays MGT, 2’s payoff is:
𝑃𝑉2 = 4 − 2𝛿 + 4𝛿 2 − 2𝛿 3 + 4𝛿 4 − 2𝛿 5 + 4𝛿 6 − 2𝛿 7 …
= 4(1 + 𝛿 2 + 𝛿 4 + 𝛿 6 + ⋯ ) − 2𝛿(1 + 𝛿 2 + 𝛿 4 + 𝛿 6 + ⋯ )
1
1
= 4 (1−𝛿2) − 2𝛿 (1−𝛿2)
=
4−2𝛿
1−𝛿 2
- Player 2’s Incentives:
1. - If player 2 defects in an odd-numbered period, his payoff is 6 in this round,
and 0 afterwards
- Player 2 has no incentive to deviate in any odd-numbered period, if:
4−2𝛿
1−𝛿 2
≥6
3𝛿 2 − 𝛿 − 1 ≥ 0
𝛿 ≥ 0.77
2. - If player 2 defects in an even-numbered period, his payoff is 0 in this round,
and 0 afterwards
- Player 2 has no incentive to deviate in any even-numbered period, if:
−2+4𝛿
1−𝛿 2
≥0
𝛿 ≥ 0.5
- Both conditions must be satisfied, therefore: 𝛿 ≥ 0.77
- How About Player 1?
- We can perform the same kind oof analysis to find that player 1 will conform to
the MGT as long as 𝛿 ≥ 0.26
- Player 1’s Incentives:
1. - If player 1 defects in an odd-numbered period, his payoff is 6 in this round,
and 0 afterwards
- Player 1 has no incentive to deviate in any odd-numbered period, if:
4+6𝛿
1−𝛿 2
≥6
3𝛿 2 + 3𝛿 − 1 ≥ 0
𝛿≥
−3+√21
6
𝛿 ≥ 0.26
2. - Player 1 has no incentive to deviate in any even-numbered period
- Both conditions for players 1 and 2 must be satisfied
- Therefore, MGT can be supported as SPNE if 𝜹 ≥ 𝟎. 𝟕𝟕
Equilibrium Payoff Set with Discounting
- Depending on the discount factor, there are many SPNE in the repeated prisoners’ dilemma
games
- (D, D) in every period (ie. Nash Equilibrium)
- (GT, GT)
- (TFT, TFT) etc.
Possible Repeated Game Payoffs: Per Period
- Any payoff inside or on the edges of the diamond can be obtained as an average payoff (ie. by
multiplying the discounted sum payoff by 1 − 𝛿) if players choose the right sequence of actions
over time
- ie. (5,1): possible if the players alternate between (C, C) and (D, C)
Equilibrium Per-Period Payoffs
- Any point on the edges or interior of the shaded area can be supported as an equilibrium
average per-period payoff, as long as the players are patient enough
Folk Theorems
- The Nash-Threat Folk Theorem:
- For repeated games with stage game G, for any feasible payoffs (M) greater than or
equal to the Nash equilibrium payoffs, and for sufficiently large discount factor, there is
a SPNE that has payoffs M
Chapter 23: Collusion, Trade Agreements, and Goodwill
- No lecture notes, see textbook
Chapters 24, 26-27: Incomplete Information
Module (Chapters 24, 26-27) Outline
- Random Events and Incomplete Information (Chapter 24)
- Bayesian Nash Equilibrium, and Applications (Chapters 26-27)
Chapter 24: Random Events and Incomplete Information
Incomplete Information Examples
- Online Auctions:
- Unrealistic to assume that bidders know the other bidders’ valuations or risk attitudes
- Oligopoly:
- Unrealistic to assume that one firm knows the cost structure of the other firm
Solution Concepts: A Comparison
Normal Form Games
Complete Information
Nash Equilibrium (NE)
Incomplete Information
Bayesian Nash Equilibrium
(BNE)
Extensive Form Games
Subgame Perfect Nash
Equilibrium (SPNE)
Perfect Bayesian Equilibrium
- Example: The Gift Game
- Chance Node: nature’s decision node
- Nature determines player 1’s type: Friend (with probability 𝑝) or enemy (1 − 𝑝)
- Player 1 observes Nature’s move, so he knows his own type
- Player 2 does not observe player 1’s type
- The Gift Game in Bayesian Normal Form:
- In games of incomplete information, rational play requires a player who knows
his own type to think about what he would have done had he been another type
- Another Example: A Game of Incomplete Information
- Player 1’s payoff number 𝑥 is private information
- Player 2 knows only that 𝑥 = 12 with probability 2/3, and 𝑥 = 0 with probability 1/3
- This matrix is not the true normal form of the game because player 1 observes 𝑥
before making his decision
- Extensive Form Representation
- Normal Form Representation
- Player 1’s decision:
1. Whether to select A or B after observing 𝑥 = 0
2. Whether to select A or B after observing 𝑥 = 12
Chapter 26: Bayesian Nash Equilibrium and Rationalizability
Finding Bayesian Nash Equilibrium (BNE)
- Method 1:
- Write down Bayesian normal form
- Solve for Nash Equilibrium of the normal form: Bayesian Nash Equilibrium
- Or solve for the set of strategies which survive iterated elimination of dominated
strategies: Bayesian Rationalizability
- Method 2:
- Treat types of each player as separate players
- We only focus on method 1
- A Game of Incomplete Information
- Player 1’s payoff number 𝑥 is private information
- Player 2 knows only that 𝑥 = 12 with probability 2/3 and 𝑥 = 0 with probability 1/3
- This matrix is not the true normal form of the game because player 1 observes 𝑥
before making his decision
- Extensive Form Representation
- Normal Form Representation
- Iterated elimination of dominated strategies:
1. 𝐵12 𝐵 0 dominates 𝐵12 𝐴0 ; 𝐴12 𝐵 0 dominates 𝐴12 𝐴0
2. 𝐷 dominates 𝐶
3. 𝐵12 𝐵 0 dominates 𝐴12 𝐵 0
{𝐵12 𝐵0 , 𝐷} is the Bayesian rationalizable set, and the unique BNE
Chapter 27: Lemons, Auctions, and Information Aggregation
Adverse Selection
- This is a problem of hidden characteristics (when one side of a transaction knows something
about itself that the other does not) and self-selection
- The uninformed party gets exactly the wrong people trading with it, so we say that the
uninformed party gets an adverse selection of the informed parties
- Example: Health Insurance Market
- Asymmetric Information:
- Situation in which one party engaged in an economic transaction has better
information than the other party
- An individual knows her own illness risk, but insurer does not
- Results in Adverse Selection:
- The phenomenon under which the uninformed side of a deal gets exactly the
wrong people trading with it
- In charging everyone the same premium, high-risk individuals have a higher
probability of buying while low-risk individuals do not
- Unhealthy people are more likely to want insurance → the proportion of unhealthy
people in the pool of insured people increases
- The price of insurance to rise → more healthy people, aware of their low risks, elect
not to be insured
- This further increases the proportion of unhealthy people among the insured, thus
forcing the price of insurance up more
- The process continues until most people who want to buy insurance are unhealthy. At
that point, insurance becomes very expensive, or – in the extreme – insurance
companies stop selling the insurance
- Used Cars and Adverse Selection
- Low-quality products can crowd out high-quality products
- There is a “market failure” because sellers of low-quality “lemons” impose a negative
externality on sellers of high-quality products
- When low-quality products are offered for sale, they adversely affect the perceived
value of high-quality products if buyers cannot differentiate low- and high-quality
- Low-quality products prevent the market for high-quality products from functioning
properly
Lemons
- Adverse selection is also know as the lemons problem
- Suppose you purchase a new car, drive it for 3 months, and then for some reasons, you must
sell it. What price do you think you could get for your car? Why?
- Used cars sell for much less than new cars because there is asymmetric information
about their quality
- The seller of the used car knows much more about the car than the prospective buyer
does
- How could you minimize the problem?
- Lemon: An Example
- Jerry is in the market for a used car, and Freddie offers an attractive 15-year-old sedan
for sale
- Assume the suggested market value for this car is 𝑷
- The car is a peach with probability 𝒒
- If peach: worth $3000 to Jerry, $2000 to Freddie
- If lemon: worth $1000 to Jerry, 0 to Freddie
- What is the efficient outcome?
- Extensive Form:
- Nature moves first
- Jerry and Freddie then choose their strategies simultaneously
- Bayesian Normal Form:
- How many strategies does Jerry have?
- How many strategies does Freddie have?
- What is the size of the matrix?
- BNE 1: Only Lemons Traded
- (𝑻, 𝑵𝑷 𝑻𝑳 ): two conditions should hold:
1. Jerry: 𝑞 ∙) + (1000 − 𝑝)(1 − 𝑞) ≥ 0 or 1000 ≥ 𝑝
2. 2000𝑞 + (1 − 𝑞)𝑝 ≥ max {𝑝, 𝑝𝑞, 2000𝑞}
- Intuition:
- If price is below $1000, Freddie would only want to bring lemons
to the market
- Anticipating that only a lemon will be for sale, Jerry is willing to
pay no more than $1000
- BNE 2: Both Lemon and Peach Traded
- (𝑻, 𝑻𝑷 𝑻𝑳 ): two conditions should hold:
1. 2000𝑞 + 1000 − 𝑝 ≥ 0 or 2000𝑞 + 1000 ≥ 𝑝
2. 𝑝 ≥ max{𝑝𝑞, 2000𝑞 + (1 − 𝑞)𝑝, 2000𝑞} or 𝑝 ≥ 2000
𝟏
- Combining both conditions: 𝒒 ≥ 𝟐
- Intuition:
1. Jerry’s expected value from owning the car exceeds its price
2. Freddie is willing to bring a peach to the market
3. The probability of a peach should be sufficiently high
- Solving the Adverse Selection Problem
- Some limited ways to address this:
- Have a mechanic check over the car
- Offer a warranty
- Establish a reputation
- Some Cures for Adverse Selection in Providing Health Care
1. Provide medical policies to entire groups (ie. through employers)
2. Make coverage mandatory
3. Refuse coverage for “pre-existing conditions”