Information, Control and Games

1
Information, Control and Games

• Shi-Chung Chang
• EE-II 245, Tel [phone] ext. 245
• scchang_at_cc.ee.ntu.edu.tw, http//recipe.ee.ntu.edu
  .tw/scc.htm
  
• Office Hours Mon/Wed 100-200 pm or by
  appointment
  
• Yi-Nung Yang
• (03 ) 2655201 ext. 5205, yinung_at_cycu.edu.tw

2
Normal Form (one-shot) games. Solution concepts
the Nash Equilibrium.

• Lecture 2

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What is a game?

• A finite set N of players
• N 1, 2, . , i , n
• A set of strategies Si for each player
• Strategies (actions ) set
• Si si si is a strategy available to
  player i Si may be finite or infinite.
  
• A payoff function ?i for each player.
• ?i assigns a payoff to player i depending on
  which strategies the players have chosen.

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Example 1

• Working on a joint project
• ????? term project
• Both work hard
• One works hard but the other goofs off
• Both goof off

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Example 1 (cont.)

• Working on a joint project
• A finite set N of players
• N 1, 2
• A set of strategies Si for each player
• Strategies (actions ) set
• Si work hard, goof off
• A payoff function ?i for each player.
• ?1(W, W) 2 ?2(W,W)
• ?1(W, G) 0 ?2(G,W)
• ?1(G, W) 3 ?2(W,G)
• ?1(G, G) 1 ?2(G,G)

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Example 1 (cont.)

• Normal Form

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Example 2

• Duopoly
• ??? ?? vs ??

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Example 3

• Coordination gameBattle of the Sexes (BoS)
• ????, ?????

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Solution to The famous Prisoners Dilemma

• Prisoners Dilemmaan engineers version
• Suppose each of two engineers wants to build a
  bridge or a tunnel across the Amazon from city A
  to city B.
  
• It costs 20 million to build a bridge and 50
  million to build a tunnel.
  
• Revenue
• If both build a bridge/tunnel, each can sell her
  bridge/tunnel for 80 million.
  
• if one builds a bridge and one builds a tunnel,
  the bridge will sell for 25 million and the
  tunnel will sell for 120 million
  
• Why? Due to high winds and heavy rains in the
  area, most people when given a choice will choose
  to drive through a tunnel.

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N 1, 2 S1 bridge, tunnel S2 ?I (brid
ge, bridge) 80-2060 ?2 (bridge, bridge)
?I (tunnel, tunnel) 80-5030 ?2 (tunnel, tunn
el) ?I (bridge, tunnel) 25-205 ?2 (tunnel,
bridge) ?I (tunnel, bridge) 120-5070 ?2 (br
idge, tunnel)x Bimatrix Form
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Solution Concepts

• What is a solution to a game?
• We want a solution to predict what strategies
  players will choose.
  
• Note solutions can also be prescriptivethey
  can tell us what strategies players should play.
  
  
• We will concentrate for now on the predictive
  performance of a solution.
  
• We can test a solutions predictive ability
  experimentally, by having subjects (often
  students) play games in a laboratory or
  empirically, by seeing how firms behave in a
  market, or how politicians behave in an election.

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The Premier solution concept The Nash
Equilibrium

• We will use the PD game to introduce the
  concept.
  
• Why is this the premier concept in game theory?
  
  
• Because it has performed relatively well in
  experimental tests and empirical tests and is
  widely applicable. (See Osbornes discussion
  p.25).

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• Lets look back at our bimatrix form of
  Prisoners Dilemma.
  
• Our two engineers see the 60, 60 payoff and would
  like to agree to build bridges.
  
• However, even if they meet to talk things over
  and agree--- Engineer 1 will Reason as follows
  If 2 builds a bridge I can earn 60 or defect to a
  tunnel and earn 70. And if 2 cheats and builds a
  tunnel, I will earn 5 or defect and earn 30.
• So no matter what 2 does I do better building a
  tunnel!!!
  
• Therefore I build a tunnel
• Player I reasons similarly and builds a tunnel.
  Each earns 30.

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We have two solution concepts so far

• Players agree to jointly optimize (bridge,
  bridge)this is also called a Pareto optimal
  outcome
  
• Dominant strategy equilibrium (tunnel, tunnel)
• Why would we predict that the dominant strategy
  will be played and not the joint optimal
  solution?
  
• The logic to playing the dominant strategy
  equilibrium is overwhelming. I earn more with
  tunnel no matter what my opponent does.
  
• Moreover, dominant strategy equilibrium tests
  well in experiments in the lab even in Prisoners
  Dilemma (see Osbornes discussion).
  
• All sorts of examples too. OPEC, anti-trust cases
  (firms found to have cheated on price agreements,
  avoidance of PD etc)..

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Pareto Optimal Outcome?

• When will the players play the joint optimal
  solution?
  
• When it is possible for players to form legal
  binding commitments. For example, write a legal
  contract.
  
• Conclusion
• On the day of the press conference the two
  engineers announce what each will build.
  
• They may have agreed before hand to build
  bridges.
  
• But they will both announce TUNNEL
• Unless they were able to write an enforceable
  contract.
  
• Not so easy to do in most situations and often
  illegal.
  
• Firms have to make their way around Prisoners
  Dilemma!

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• Comments
• 1. We are discussing noncooperative game theory
  where no binding contracts are
  
• allowed.
• We are discussing one-shot simultaneous play
  games where both players must
  
• announce their strategies simultaneously
  and the game is played once.
  
• We have seen an example of a dominant strategy
  equilibrium. Heres the
  
• definition for a two player game.
• A dominant strategy equilibrium is a strategy
  pair (s1, s2) such that s1 ? S1,
  
• s2 ? S2 ,
• ?I (s1, s2) ?I (s1, s2) for all s1 ? S1, s2 ?
  S2 ,and s1 not equal to s1
  
• ?2 (s1, s2) ?2(s1, s2) for all s1 ? S1, s2 ?
  S2 ,and s2 not equal to s2.
  
• Most games dont have dominant strategy
  equilibrium. Thats why Nash
  

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Definition

• A Nash equilibrium is an strategy profile s with
  the property that no player i can do better by
  choosing and action different from si, given
  that every other player j adheres to sjs
  si, sj

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Now lets define a Nash equilibrium.
We will look at a game that has a Nash equilibri
um, but no dominant strategy Equilibrium. Her
e are four equivalent definitions of a Nash
equilibrium. First two give us a feeling for wh
at a Nash equilibrium is. The second two are us
eful for funding the Nash equilibrium or
equilibria for a specific game.
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Given a game G (N 1,2 S1, S2 ?I, ?2),
the strategy pair (s1, s2) is a Nash equilibri
um for G if 1.Neither player has an incentive t
o unilaterally defect to another strategy.
2. s1 is a best response to s2 and s2 is a
best response to s1. 3. ?1 (s1, s2) ?
?1 (s1, s2) for all s1 ? S1.
and ?2 (s1, s2) ? ?2 (s1, s2) for all s2
? S2. 4. ?I (s1, s2) is a column maxim
um and ?2 (s1, s2) is a row maximum.
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Uniqueness of Nash equil.

• If a Nash equil. exist, is it unique?
• Example Battle of the sexes
• It is a Saturday night, Geroge loves to watch
  football, but Marry enjoys opera....
  
• They also like each others company...
• Find the Nsah equil.?

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Existence of Nash equilibrium

• Same as before with a slight modification
• George wants to meet Marry. However, Marry wants
  to avoid George
  
• The only activities are a movie and a dance
• Marry prefers to be alone, but if she must be
  with George, she prefers the movie, since she
  wont have to talk to George.
  
• George prefers to be with Marry, and if he
  succeeds, he prefers the dance, where he can talk
  to her.
  
• Find Nash eq.

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Exercise A Sealed Bid Auction. Suppose two
bidders bid for an item they know they can sell
for 20. The rules of the auction require a b
id of 16, 10, or 4. If both bidders submit
the same bid, they share the item.
Put the game in normal form.
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2
H M
L
2,2 4,0 4,0
H
0,4 5,5
10, 0
1
M
0,4 0, 10
8,8
L
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• 1. Is there a dominant strategy equilibrium?
• What is player 1s best response to H?
• What is player 1s best response to M?
• What is player 1s best response to L?
• So no one strategy of player 1 is a best response
  to all strategies of player 2.
  
• Find all Nash equilibria.
• Is (H,H) a Nash equilibrium?
• Is (H,M) a Nash equilibrium?
• etc.
• Note a Nash equilibrium is a strategy profile and
  should not be given in terms of
  
• payoffs. ?? Nash equil. ??????, ?????

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Strict and nonstrict equilibria
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Dominant Strategy

• For player 1
• T is dominated by M
• T is dominated by M
• M is dominated by B

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Cournot Game

• Cournots duopoly game
• Two firms produce identical products and competes
  in a market
  
• Market demand P(Q) P(q1 q2), P' (Q)
• Each firms profit Revenue - Cost ?i (qi, q-i)
  P(qi q-i) qi - Ci(qi)
  
• Optimization maximizing profitsFOC ??i(qi,
  q-i)/?qi P'(qi q-i) qi P - C'i ? 0, for i1,
  2
  
• Best response function (reaction curve)qi
  qi(q-i)
  
• Solve q1, q2 simultaneously to yield Nash solution

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Cournot Game an example

• Market DemandPP(Q) ? - Q, Q q1 q2
• Common Constant Marginal CostCi c qi , for i
  1,2
  
• Profits ?i (qi, q-i) (? - qi- q-i)qi - cqi
• FOCResponse function??i (qi, q-i)/?qi -qi(?
  -qi- q-i -c) ? 0

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Profit Function in Cournot Game

• Profit function?1q1(? -q1- q2 -c)given any
  q2when q2 0 ?1 q1(? -q1-c) q1 0,
  ?-cwhen q2 0Profit curve shifts downward
  ?1 q1(? -q1 - q2 -c)

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Best Response f() and Nash

• Firm is optimal choice of qi given others q-i
  
  
• For firm 1, FOC becomes(-1)q1(? -q1- q2 -c) ?
  0q1 (1/2) (? -q2 -c)
  
• For firm 2, FOC becomes -q2(? -q1- q2 -c) ?
  0q2 (1/2) (? -q1 -c)
  
• Cournot-Nash equilibriumqi (1/3) (? -c), for
  i 1, 2

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Reaction Curves and Nash
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Nash Equil. In Cournot Game
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A Collusive Duopoly Outcome

• Two firms collude as a monopoly
• They maximize joint profits and share the output
• Market DemandPP(Q) ? - Q,
• Joint Profits max ? P(Q)Q - cQ (? - Q)Q -
  cQ
  
• FOC (? - Q) -Q - c 0 Qm q1q2(?-c) /2?
  Each firms collusive output qim (?-c) /4
  
  
• OPEC collusion

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A Collusive Duopoly Outcome is not a Nash
equilibrium?
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Bertrands Competition

• Price (cost) competition
• Firms set prices to maximize profits
• Consumers purchase with the lowest price
• A Firm takes ALL with the lowest price. Firms
  share the market equally if prices are the same
  
• The Game
• Player the firms (with cost function Ci(qi)
• Strategies each firms possible (non-negative)
  prices
  
• Payoffs for firm i (market demand D ? -
  p)piD(pi) / m - Ci(D(pi)/m)if there are m firms
  with the same lowest price,where m 1 if firm
  is profits is lower than the others

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Profit function in Bertrand Game

• Bertrands duopoly game
• Two firms compete in the market

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Profits in Duopoly Bertrand

• when pj profit 0 if pipj
  
• Best responseBi(pj)pi pipj
• when pj c, similar to the aboveprofit 0 if
  pi?pj

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Profits in Duopoly Bertrand (2)

• when c piprofit 0 if pipj
  
• Best response seems to beempty set

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Profits in Duopoly Bertrand (3)

• when pj pm, firm is best responseBi(pj)pi
  pipm

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Best Response f() in Bertrand
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Best Response Plot in Bertrand

• Nash equilibrium (p1, p2) (c, c)

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Reasoning in Bertrand

• No one should set pi feasible strategy set is pi ? c, for i1,2
  
• If firm i choose pi lower pj to take All market.
  
• But firm i also does the same thing. So the price
  continued to be lower (price war) until pi c.
  
• Zero-profit Nash outcome
• zero profit normal profit