Title: Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory with Complete Information
1Introduction to Game Theory (Deterministic Model)
Non-cooperative Game Theorywith Complete
Information
- Prof. Dr. Jinxing Xie
- Department of Mathematical Sciences
- Tsinghua University, Beijing 100084, China
- http//faculty.math.tsinghua.edu.cn/jxie
- Email jxie_at_ math.tsinghua.edu.cn
- Voice (86-10)62787812 Fax (86-10)62785847
- Office Rm. 1202, New Science Building
2What is Game Theory?
- No man is an island
- Study of rational behavior
- in interactive or interdependent situations
- Bad news
- Knowing game theory does not guarantee winning
- Good news
- Framework for thinking about strategic
interaction
3Games We Play
- Group projects free-riding, reputation
- Flat tire coordination
- GPA trap prisoners dilemma
- Tennis / Baseball mixed strategies
- Mean professors commitment
- Traffic congestion
- Dating information manipulation
4Games Businesses Play
- Patent races game of chicken
- Drug testing mixed strategies
- FCC spectrum auctions
- Market entry commitment
- OPEC output choice collusion enforcement
- Stock options compensation schemes
- Internet pricing market design
5Why Study Game Theory?
- Because the press tells us to
- As for the firms that want to get their hands on
a sliver of the airwaves, their best bet is to go
out first and hire themselves a good game
theorist. - The Economist, July 23,1994 p. 70
- Game Theory, long an intellectual pastime, came
into its own as a business tool. - Forbes, July 3, 1995, p. 62.
-
- Game theory is hot.
- The Wall Street Journal, 13 February 1995,
p. A14
6Why Study Game Theory?
- Because we can formulate effective strategy
- Because we can predict the outcome of strategic
situations - Because we can select or design the best game for
us to be playing
7Why Study Game Theory?
- McKinsey
- John Stuckey David White - Sydney
- To help predict competitor behavior and
determine optimal strategy, our consulting teams
use techniques such as pay-off matrices and
competitive games. - Tom Copeland - Director of Corporate Finance
- Game theory can explain why oligopolies tend to
be unprofitable, the cycle of over capacity and
overbuilding, and the tendency to execute real
options earlier than optimal.
8Outline -- Concepts
- Recognizing the game
- Rules of the game
- Simultaneous games
- Anticipating rivals moves
- Sequential games
- Looking forward reasoning back
- Mixed strategies
- Sensibility of being unpredictable
- Repeated games
- Cooperation and agreeing to agree
9Outline -- Applications
- Winning the game
- Commitment
- Credibility, threats, and promises
- Information
- Strategic use of information
- Bargaining
- Gaining the upper hand in negotiation
- Auctions
- Design and Participation
10Interactive Decision Theory
- Decision theory
- You are self-interested and selfish
- Game theory
- So is everyone else
- If its true that we are here to help others,
- then what exactly are the others here for?
- - George Carlin
11The Golden Rule
COMMANDMENT Never assume that your opponents
behavior is fixed. Predict their reaction to your
behavior.
12The Matrix of Game Theory
13The Matrix of Non-cooperation Game
Complete (Full) Information Incomplete Information
Simultaneous Move (Static) Nash Equilibrium Bayesian Equilibrium
Sequential Move (Dynamic) Subgame Perfect (Nash) Equilibrium Subgame Perfect Bayesian Equilibrium
14Definition of a Game
- Must consider the strategic (Normal) environment
- Who are the PLAYERS? (Decision makers)
- What STRATEGIES are available? (Feasible
actions) - What are the PAYOFFS? (Objectives)
- Rules of the game
- What is the time-frame for decisions?
- What is the nature of the conflict?
- What is the nature of interaction?
- What information is available?
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16The Assumptions
- Rationality
- Players aim to maximize their payoffs
- Players are perfect calculators
- Common knowledge
- Each player knows the rules of the game
- Each player knows that each player knows the
rules - Each player knows that each player knows that
- each player knows the rules
- Each player knows that each player knows that
each player knows that - each player knows the rules
- Each player knows that each player knows that
each player knows that each player knows that
each player knows the rules - Etc. etc. etc.
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29Nash (1950)
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34Cournots Model of Oligopoly(Cournot, 1838)
- Single good produced by n firms
- Cost to firm i of producing qi units Ci(qi),
where Ci is nonnegative and increasing - If firms total output is Q then market price is
P(Q), where P is nonincreasing - Profit of firm i, as a function of all the firms
outputs
35Cournots Model of Oligopoly
- Strategic (normal) form game
- players firms
- each firms set of actions set of all possible
outputs - each firms preferences are represented by its
profit
36Example Duopoly
- constant unit cost Ci(qi) cqi, where c lt a
37Example Duopoly
38Example Duopoly
39Example Duopoly
- Best response function is
Same for firm 2 b2(q) b1(q) for all q.
40Example Duopoly
41Example Duopoly
- Nash equilibrium
- Pair (q1, q2) of outputs such that each firms
action is a best response to the other firms
action - or
- q1 b1(q2) and q2 b2(q1)
- Solution
- q1 ( a - c - q2)/2 and q2 (a - c - q1)/2
- q1 q2 (a - c)/3
42Example Duopoly
43Example Duopoly
- Conclusion
- Game has unique Nash equilibrium
- (q1, q2) (( a - c)/3, (a - c)/3)
- At equilibrium, P (a 2c)/3, each firms
profit is - p ((a - c)2)/9
- Total output 2(a - c)/3 lies between monopoly
output (a - c)/2 and competitive output a - c.
44Cournots Model of Oligopoly Notes
- Dependence of Nash equilibrium on number of firms
- Comparison of Nash equilibrium with collusive
outcomes (monopoly) - If there are only one firm in the
market - Max (a-q-c)q ? q (a-c)/2 lt
2(a-c)/3 - If P(Q) a-bQ Cournots model gives
- (q1, q2) (( a - c)/3b, (a -
c)/3b) - P (a 2c)/3
(unchanged) - p ((a - c)2)/9b
45Bertrands Model of Oligopoly (Bertrand, 1883)
- Strategic variable price rather than output.
- Single good produced by n firms
- Cost to firm i of producing qi units Ci(qi),
where Ci is nonnegative and increasing - If price is p, demand is D(p)
- Consumers buy from firm with lowest price
- Firms produce what is demanded
46Bertrands Model of Oligopoly(Bertrand, 1883)
- Strategic game
- players firms
- each firms set of actions set of all possible
prices - each firms preferences are represented by its
profit
47Example Duopoly
- 2 firms
- Ci(qi) c qi for i 1, 2
- D(p) a - p ? p a - D
- P in 0, 8, or actually in 0, a?
48Example Duopoly
- Nash Equilibrium
- (p1, p2) (c, c)
- total quantity produced a - c (?)
- If each firm charges a price of c then the other
firm can do no better than charge a price of c
also (if it raises its price it sells no output,
while if it lowers its price it makes a loss), so
(c, c) is a Nash equilibrium.
49Example Duopoly
- No other pair (p1, p2) is a Nash equilibrium
since - If pi lt c then the firm whose price is lowest (or
either firm, if the prices are the same) can
increase its profit (to zero) by raising its
price to c - If pi c and pj gt c then firm i is better off
increasing its price slightly - if pi pj gt c then firm i can increase its
profit by lowering pi to some price between c and
pj (e.g. to slightly below pj if D(pj) gt 0 or to
pmonop if pj gt pmonop).
50Bertrands Model Notes
- If D(p) has the form p a - bD
- Nash Equilibrium unchanged (p1, p2) (c, c)
- total quantity produced (a - c)/b (0?)
- If the products produced by two firms are
non-identical Di(pi) a - pi bpj (i2-j) - ? pi( a c bpj)
- ? p1p2(a c ) / (2-b)
51Hotellings Model of Electoral Competition
- Several candidates run for political office
- Each candidate chooses a policy position
- Each citizen, who has preferences over policy
positions, votes for one of the candidates - Candidate who obtains the most votes wins.
52Hotellings Model of Electoral Competition
- Strategic game
- Players candidates
- Set of actions of each candidate set of possible
positions - Each candidate gets the votes of all citizens who
prefer her position to the other candidates
positions each candidate prefers to win than to
tie than to lose. - Note Citizens are not players in this game.
53Example
- Two candidates
- Set of possible positions is a (one-dimensional)
interval. - Each voter has a single favorite position, on
each side of which her distaste for other
positions increases equally. - Unique median favorite position m among the
voters the favorite positions of half of the
voters are at most m, and the favorite positions
of the other half of the voters are at least m.
54Example
- Direct argument for Nash equilibrium
- (m, m) is an equilibrium if either candidate
chooses a different position she loses. - No other pair of positions is a Nash equilibrium
- If one candidate loses then she can do better by
moving to m (where she either wins or ties for
first place) - If the candidates tie (because their positions
are either the same or symmetric about m), then
either candidate can do better by moving to m,
where she wins.
55Sequential Game
- Life must be understood backward,
but it must be lived forward. - - Soren Kierkegaard
-
56Games of Chicken
- A monopolist faces a potential entrant
- Monopolist can accommodate or fight
- Potential entrant can enter or stay out
Monopolist
Accommodate Fight
In 50 , 50 -50 , -50
Out 0 , 100 0 , 100
Potential Entrant
57Equilibrium
- Use best reply method
- to find equilibria
Monopolist
Accommodate Fight
In 50 , 50 -50 , -50
Out 0 , 100 0 , 100
Potential Entrant
58Importance of Order
- Two equilibria exist
- ( In, Accommodate )
- ( Out, Fight )
- Only one makes temporal sense
- Fight is a threat, but not credible
- Not sequentially rational
- Simultaneous outcomes may not make sense for
sequential games.
59Sequential Games
The Extensive Form
60Looking Forward
- Entrant makes the first move
- Must consider how monopolist will respond
- If enter
- Monopolist accommodates
61 And Reasoning Back
- Now consider entrants move
- Only ( In, Accommodate ) is sequentially rational
62Sequential Rationality
COMMANDMENT Look forward and reason
back. Anticipate what your rivals will do
tomorrow in response to your actions today
63Solving Sequential GamesBackward Induction
- Start with the last move in the game
- Determine what that player will do
- Trim the tree
- Eliminate the dominated strategies
- This results in a simpler game
- Repeat the procedure
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73Equilibrium
- What is likely to happen
- when rational players interact in a game?
- Type of equilibrium depends on the game
- Simultaneous or sequential
- Perfect or limited information
- Concept always the same
- Each player is playing the best response to
other players actions - No unilateral motive to change
- Self-enforcing
74Summary
- Recognizing that you are in a game
- Identifying players, strategies, payoffs
- Understanding the rules
- Manipulating the rules
- Nash Equilibrium
- Subgame Perfect Equilibrium
- Best response strategy (function)
- Backward Induction
- Searching for possible outcomes