Dynamic Games of Complete Information

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Dynamic Games of Complete Information

• .

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Extensive form games

• To model games with a dynamic structure
• Main issues with a dynamic structure
• 1. Information structure who knows what and
  when?
• 2. Credibility
• 3. Commitment
• 4. The idea of Backward Induction

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The Stackelberg game

• A dynamic version of the Cournot game
• Player 1, Stackelberg leader chooses output q1
  first
• Player 2, Stackelberg follower chooses output
  q2 next
• Demand is linear p(q)12-q
• Player is utility is ui(q1 , q2 )12- (q1
  q2)qi
• What is the Stackelberg equilibrium?
• Are there other Nash equilibria?

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Model of strategic investment

• Firms 1, 2 have average cost of 2 per unit
• Firm 1 can install new technology at cost f. Then
  average cost is zero
• Firm 2 can observe firm 1s investment
• The two firms then move simultaneously to set
  quantity. Demand is p(q)14-q
• How should firm 1 forecast its rivals output?
• Backward induction not directly applicable. Why?

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The Extensive form

• Building blocks of the extensive form game
• 1. The set of players
• 2. The order of moves - i.e. who moves when
• 3. The players payoffs as a function of moves
• 4. What the players choices are when they move
• 5. What a player knows when making his choice
• 6. Probability distribution over any exogenous
  events

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Extensive form trees

• Rules for forming trees
• 1. Single starting point
• 2. No cycles
• 3. One way to proceed
• Define precedence relation a b a precedes
  b
• 1. is asymmetric a b, means b a
• 2. is transitive
• 3. x/ x and x// x implies x/ x//
  or x// x/
• 4. There is single initial node

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Formalizing the extensive form

• 1. Let i ?I be the finite set of players
• 2. Let i(x) bet set of players that move at node
  x
• 3. Let Z be set of terminal nodes. Maps uiZ?R
  with values ui(z) are is payoffs to a sequence
  of moves z
• 4. Let A(x) be set of feasible actions at node x
• 5. Information Sets h partition nodes of the
  tree
• a. Each node x is in only one information set
  h(x)
• b. If x/?h(x), then player moving at x does not
  know if he is at x or x/
• c. If x/?h(x), then the same player moves at x
  x/
• d. If x/?h(x), then A(x) A(x/). Thus A(h) is
  action set at information set h

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Example

• Two people want to go to a Broadway musical in
  great demand
• There is exactly one ticket left, and whoever
  arrives first gets it
• There are three transportation choices c(cab)
  b(bus) s(subway)
• Player 1 leaves home a little earlier
• A cab is faster than the subway, which is faster
  than a bus

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Strategies equilibria in extensive form

• Let Hi be set of player is information sets
• Let be the set of all actions for i
• A pure strategy for i is a map si Hi ? Ai , with
  
• si(hi) ? A(hi) for all hi ? Hi
• The set of pure strategies for i is Si
• The number of is pure strategies is given by the
  product
• Mixed strategies in extensive form are called
  behavior strategies. Let ?(A(hi)) be prob dist on
  A(hi)
• A behavior strategy for i, denoted bi, is an
  element of Cartesian product

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Strategic-form versus extensive-form

• Using its pure strategies and payoffs, an
  extensive form can be transformed to strategic
  form
• Extensive form interpretation player i waits
  until hi is reached before deciding how to play
  there
• Strategic form interpretation player i makes a
  complete contingent plan in advance
• Games of perfect information with all singleton
  information sets constitute a special class
• Any mixed strategy si (strat form) generates a
  behavior strategy bi (ext form), but many
  different sis can generate the same bi
• Theorem (Kuhn 1953)
• In a game of perfect recall, mixed and behavior
  strategies are equivalent

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Backward induction Subgame perfection

• Theorem (Zermelo 1913 Kuhn 1953)
• A finite game of perfect information has a pure
  strategy Nash equilibrium
• Subgame perfection is the analog of backward
  induction for multi-player situations
• G is a proper subgame of an extensive form game T
  if it
• 1. Starts at a single node x of T
• 2. Contains all successors of x
• 3. If x/? G, and x//? h(x/), then x//? G
• A behavior strategy s of an extensive form game
  is a subgame perfect equilibrium if the
  restriction of s to G is a Nash equilibrium of G
  for every proper subgame G

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Multi-stage games with observed actions

• 1. There are k stages 0, 1, , k-1
• 2. All players know the actions chosen at all
  previous stages
• 3. All players move simultaneously in each stage
• 4. This includes games where players move
  alternately (all other players have strategy do
  nothing)

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Multi-stage games with observed actions

• Let a0 be the stage-0 action-profile
• At the beginning of stage1, players know history
  h1 which is just a0
• Let Ai(h1) be player is action set at stage 1
  with history h1
• hk1 is history at end of stage k, hk1(a0,
  a1, ak), and Ai(hk1) is player is action set
  at stage k1
• If game is K stages, HK is set of all terminal
  histories
• A pure strategy for i is seq. of maps such that
  
• where

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Multi-stage games with observed actions

• Payoffs are defined on terminal histories,
• ui Hk1?R
• In most applications, payoffs are additively
  separable over stages. This isnt necessary
• The game from stage k on with history hk is a
  proper subgame G(hk), and a strategy profile s
  for whole game induces sihk for subgame G(hk)
• A Nash equilibrium s satisfies the familiar
  condition
• ui(si , s-i) ui(s/i , s-i) for all s/i
• A Nash equilibrium s is subgame perfect if sihk
  is a Nash equilibrium for every subgame G(hk)

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Principle of optimality and subgame perfection

• For multi-stage games with observed actions, we
  have a useful characterization of subgame
  perfection for the finite-horizon case
• Theorem (One-stage-deviation principle)
• A strategy profile s is subgame perfect iff no
  player i can gain by deviating from s in a
  single stage and conforming to s thereafter
• This theorem can be extended to the infinite
  horizon case

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Rubinstein Bargaining model

• Two players have to share a pie of size 1
• The game
• Step1 In periods 0, 2, 4,, player 1 proposes a
  split (x, 1-x)
• Step 2 If player 2 accepts in period 2k, game
  ends. If he rejects, he proposes (x, 1-x) in
  period 2k1
• Step 3 If player 1 accepts, game ends. Else,
  Step 1
• Discount factors are d1, d2, and if split (x,
  1-x) is accepted at time t payoffs are (dt1x,
  dt2(1-x))
• This is an infinite-horizon game of perfect info

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Subgame perfect equilibrium

• A continuation payoff of a strategy profile in
  subgame starting at t is utility in time-t units
  of outcome induced by that profile
• Let , be player 1a lowest and highest
  continuation payoffs in any subgame that begins
  with player 1 making an offer
• Let , be player 1a lowest and highest
  continuation payoffs in any subgame that begins
  with player 2 making an offer
• Similarly define for player 2

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Subgame perfect equilibrium

• When I makes offer, 2 will accept if he gets more
  than d2 . Hence, . Also, by symmetry,
• Suppose player 1 makes offer (x, 1-x). If 2
  accepts, the min he can get is , and
  therefore, 1-x . Thus, x1- . This
  implies that, ,
• Again, by symmetry,

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Subgame perfect equilibrium

• Now, , so,
• Similarly, , which gives,
• But, , and thus,
• Proceeding similarly,
• What is the effect of patience?
• As d1?1, for fixed d2, we have v1?1, and player
  1 gets entire pie.

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A model of RD race

• Firms R and S are conducting RD
• Several stages need to be completed
• Simplifying assumptions
• 1. Distance from goal can be measured. E.g. firm
  S is n-steps away from completion
• 2. Either firm can move 1, 2, or 3 steps
• 3. It costs 2/7/15 to move 1/2/3 steps
• 4. Firm completing all the steps first gets
  patent worth 20
• What would happen if R-S were a cartel,
  maximizing joint profits?
• - Since only one firm gets patent, only it does
  RD
• - Chosen firm moves 1-step at a time, and firm
  closer to finishing is chosen

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The extensive form of RD game

• Suppose the firms take turns deciding on RD
  investment becomes a game of perfect info
• Converting to extensive-form
• 1. Transform to location-space picture
• 2. Let (r, s) be coordinates of R and S , with r
  depicting how far R is from finishing

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Subgame perfect equilibrium of RD game

• If S is in Rs safety zone- whatever the zone
  number -it should drop out of the race
• Firm S in its own safety zone spends the minimum
  amount on RD, moves one step at a time and wins
  the patent
• In Trigger zone n, each firm spends what it needs
  to- profitably -to get an advantage and move the
  game to its safety zone n-1

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David vs Goliath in Entry Decisions

• Suppose Goliath has 700 and David has 300
• They are gambling types, and prefer roulette
• Whoever ends up with more money after the next
  round will win ultimately
• Suppose David moves first and makes the safest
  bet
• He can never win ?

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David vs Goliath in Entry Decisions

• He should take one of the more risky gambles
• Bets 300 that the ball would land on a multiple
  of 3 wins 900 w.p. 12/37
• What is Goliaths best response?
• To exactly imitate Davids bet !
• Again, David can never win ?
• Is there any hope for David?

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David vs Goliath in Entry Decisions

• David should have gone second and differentiated
  himself
• This situation is parallel to new product launch
  decisions when a firm with shallow pockets
  competes against a firm with deep pockets
• If going second is not feasible, then entrant
  should take riskier bets like launching a
  product with some chance of failing!!

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Patent races

• When is there competition or monopoly?
• Depends on possibility of preemption and
  leapfrogging
• Not about chance of winning, but about chance of
  being favorite
• Consider two firms i1,2, and let value of patent
  be V
• Productivity of RD increases over time
• Let ?i(t) be firm is total experience at time t
• µ(?i(t)) is firm is hazard rate at t. Let
  µi(t)µ(?i(t))
• Discovery probability is an exponential waiting
  time
• Note, discovery is stochastic and firm 2 (enters
  t2gt0) could discover before firm 1 (enters t1 0)

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Assumptions and preliminaries

• Cost is c and common discount rate is r
• Probability that no firm makes discovery before
  time t is
• Expected value of patent race for firm i is
• RD is viable for monopoly,
• It is unprofitable for both firms to always do
  RD
• State of competition is pair of experiences

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Model without leapfrogging

• Result 1 (?-preemption) In the unique subgame
  perfect equilibrium, whatever t2, firm 1 engages
  in RD and firm 2 drops out of the race.
• Sketch of proof
• i. Let O(?) be firm 1s experience such that it
  has zero payoff when firm 2 has experience ?, and
  both firms do RD
• ii. Show that a firm does RD until discovery or
  drops out immediately
• a) Suppose not. Let initial state be and
  both firms do RD till time t and firm 1 drops
  out at t with zero profits
• b) Then and
• c) Firm 1s expected profit at is

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Sketch of proof of Result 1 (contd.)

• d) Since for st,
  implies so firm 1 would not join
  patent race a contradiction
• iii. The strategy If ?i ?j, firm i stays in
  and j drops out iff
• ?j O(?i), is subgame perfect
• iv. Suppose there exists t such that,
  conditional on neither having dropped out at t,
  firm 1 drops out with some prob.
• v. Let be the supremum of such times for
  firm 1. for firm 2
• vi. Claim
• vii. Claim Firm 2 drops out with probability 1
  at time
• viii. Then, firm 1 will not drop out at time
  -? for small ?
• ix. Proceeding similarly, firm 1 never drops out
• x. Then, firm 2 never does RD

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Model with leapfrogging

• There are 2 stages preliminary and final stages
• Costs are c1 and c2
• Time-t experience for firm i in stage j is
  ?ji(t)
• Probability of making 1st, 2nd stage discovery at
  time t if it has not been made before are µi(t),
  ?
• Cannot accumulate 2nd stage experience without
  making 1st stage discovery
• Payoff for 2nd stage monopolist,
• (?V-c2)/(r ?), and for a 2nd stage duopolist
  is
• W2 (?V-c2)/(r 2?),

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Model with leapfrogging

• Result 2 There exists a SPNE where the leading
  firm always does RD unless the rival does RD
  and completes the 1st stage before . The
  follower either, (i) drops out at the start, (ii)
  does RD until , or (iii) always does RD
  unless the leader passes the 1st stage before (
  t2)

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Model with leapfrogging

• Sketch of proof of Result 2
• i. There are 2 decision points one firm has
  finished 1st stage, or both are in 1st stage
• ii. Let be level of 1st stage experience
  such that a firm with experience less (greater)
  than will drop out (stay in) even if the
  rival has completed stage 1. It is defined by
• iii a. If 2 has completed stage 1, firm 1 should
  stay in if t
• b. If 1 has completed stage 1, firm 2 should
  stay in if t-t2

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Model with leapfrogging

• Sketch of proof of Result 2
• iv. Suppose neither has made 1st stage discovery
  at time
• v. Firm 1 will hit experience level before
  firm 2. After that it will stay in.
• vi. Can 2 do RD profitably when both stay in
  forever? If yes, both do RD unless firm 1
  passes stage 1 before
• t2 . Else, the subgame from onwards
  resembles the no-leapfrogging situation. From
  result 1, 2 will drop at
• vii. Consider race before time .
  Straightforward extension of arguments in Result
  1 show that
• - there is ?-preemption and 2 quits at t0
• - firm 2 does RD till t t2