Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory with Complete Information - PowerPoint PPT Presentation

1 / 74
About This Presentation
Title:

Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory with Complete Information

Description:

Title: Training Last modified by: jxie Created Date: 6/2/1995 10:16:36 PM Document presentation format: Other titles: Times New Roman Arial ... – PowerPoint PPT presentation

Number of Views:2861
Avg rating:3.0/5.0
Slides: 75
Provided by: facultyMa9
Category:

less

Transcript and Presenter's Notes

Title: Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory with Complete Information


1
Introduction 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

2
What 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

3
Games 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

4
Games 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

5
Why 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

6
Why 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

7
Why 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.

8
Outline -- 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

9
Outline -- 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

10
Interactive 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

11
The Golden Rule
COMMANDMENT Never assume that your opponents
behavior is fixed. Predict their reaction to your
behavior.
12
The Matrix of Game Theory
13
The 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
14
Definition 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?

15
(No Transcript)
16
The 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.

17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
Nash (1950)
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
Cournots 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

35
Cournots 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

36
Example Duopoly
  • two firms
  • Inverse demand
  • constant unit cost Ci(qi) cqi, where c lt a

37
Example Duopoly
38
Example Duopoly
39
Example Duopoly
  • Best response function is

Same for firm 2 b2(q) b1(q) for all q.
40
Example Duopoly
41
Example 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

42
Example Duopoly
43
Example 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.

44
Cournots 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

45
Bertrands 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

46
Bertrands 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

47
Example 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?

48
Example 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.

49
Example 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).

50
Bertrands 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)

51
Hotellings 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.

52
Hotellings 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.

53
Example
  • 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.

54
Example
  • 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.

55
Sequential Game
  • Life must be understood backward,
    but it must be lived forward.
  • - Soren Kierkegaard

56
Games 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
57
Equilibrium
  • Use best reply method
  • to find equilibria

Monopolist
Accommodate Fight
In 50 , 50 -50 , -50
Out 0 , 100 0 , 100
Potential Entrant
58
Importance 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.

59
Sequential Games
The Extensive Form
60
Looking 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

62
Sequential Rationality
COMMANDMENT Look forward and reason
back. Anticipate what your rivals will do
tomorrow in response to your actions today
63
Solving 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

64
(No Transcript)
65
(No Transcript)
66
(No Transcript)
67
(No Transcript)
68
(No Transcript)
69
(No Transcript)
70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73
Equilibrium
  • 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

74
Summary
  • 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
Write a Comment
User Comments (0)
About PowerShow.com