Strategic Decisions

1
Strategic Decisions

• Class 3
• Putting Yourself in the Other
• Persons Shoes

2
Overview

• In many situations, a player will have to
  determine the best strategy at the the same time
  as a rival
• Even if the moves are not literally taking place
  at the same moment, the game is a simultaneous
  one if both rivals are unaware of the others
  move until after both have taken place

3
Common Simultaneous ID Location Game

• Two beach vendors (selling drinks, t-shirts,
  other paraphenalia) have been awarded rights to
  sell on a one mile stretch of beach. Each vendor
  may select a single site anywhere along the beach
  side of a walkway that stretches the length of
  the beach. The vendors must submit in writing
  their decision as to where they will locate by
  the first of March but they may not consult each
  other about locations. Beach season is Memorial
  Day through Labor Day.  During this period daily
  attendance fluctuates, but on both heavily and
  lightly attended days, beach-goers tend to
  distribute themselves fairly evenly along the one
  mile stretch.
• What statements can you make about best
  locations for on of locations for the two
  vendors?

4
Examples of Simultaneous IDs

• Product Marketing
• Positioning -- price-quality mix and specific
  geographic location decisions
• Pricing Wars
• Bargaining compensation, contracts,
• Negotiation over allocation of profits in a joint
  venture or merger that would benefit both
  parties. Delays or impasses reduce the size of
  the pie.
• Voting and politics
• One unit gains at other units expense

5
Analyzing Simultaneous IDs

• Two Main Kinds Constant Sum Variable Sum
• Constant Sum Games (e.g. zero sum) focus here
• Dividing a fixed pie
• Purely competitive situations
• Price decisions dividing well established market
• Variable Sum (e.g. positive sum) Games
  -- more in Class 4
• Strategies determine size of pie division
• Cooperate-compete situations
• Example Joint venture player-team negotiations

6
Analyzing Simultaneous IDs Finding Answers

• Each player puts self in other players shoes
• However, easy to fall into chasing your own
  tail
• If I do x, rival does y, but if rival does y, I
  do z,
• Need Solution Technique to find Nash
  Equilibrium
• NE A pair (or set) of strategies that are best
  responses to one another

7
Thinking about NE Dominant Dominated
Strategies

• Determine whether a players strategy is best
  regardless of what other chooses (dominant)
• Then determine best strategy for other player
• Determine whether a players strategy is worst
  regardless of what other chooses (dominated)
• Eliminate dominated options and then look for
  dominant strategies

8
Back to the Location Example

• Consider the Beach Vendor Problem
• Simplifying Problem
• 2 Vendors (1, 2) choosing at same time
• Beachgoers evenly distributed along section of
  beach
• Location Options Left, Middle, Right (L, M, R)
• No cooperation -- Fixed Pie (zero sum)
• Consider payoffs to combinations (draw map)
• NE Solution
• Extremes are dominated can always do at least
  as well in middle

9
Location Problem Solution

• NE to simple case
• Rivals side-by-side in middle of beach
• Simple case serves as common template
• Retail stores including fast-food, gas stations
  evening news show times political candidates
• Exceptions
• Why?
• What are the limits of the simplifications and
  how might changing them influence the results?

10
Simultaneous ID Tables

• In simultaneous move games, tables are often help
  clarify options
• used instead of a game trees because the sequence
  of moves no longer matters
• Rows columns corresponds to strategies for the
  players
• The cells of the table depict the payoffs for the
  row and column player respectively

11
Simultaneous ID Mechanics Pitcher-Hitter Example

• Consider 2 firms (hitter pitcher) Firms
• Pitcher chooses fastball or off-speed
• Hitter guesses fastball or off-speed
• Look for dominant strategy
• If none, look for dominated strategy
• If needed, then look again for dominant

12
ID with Dominant Strategy
13
Explaining Solution Mechanics

• Examine outcomes for each column (hitter
  strategy) then determine pitchers best choice
  for that column
• If hitter guessing FB, FB or OS is same for
  pitcher
• If hitter guessing OS, FB is pitchers best
  choice
• FB always at least as good as OS for pitcher
• FB is (weakly) dominant strategy for pitcher
• Knowing this, hitter chooses from FB row
• Guessing FB is best option
• FB (pitcher) FB (hitters) is Nash Equilibrium
• If not dominant strategy for player 1, repeat the
  procedure above for player 2

14
Expanding the Situation

• Same 2 firms pitcher hitter
• Now, 3 decision possibilities for each
• Fastball, change-up, slider
• Advance warning
• No dominant solution using mechanics just
  described
• Proceed to looking for dominated solutions

15
ID with Dominated Strategy
16
Dominated Strategies

• Iterate through pitchers choices
• If hitter guessing FB, pitchers best option is
  CU
• If hitter guessing CU, pitchers best option is
  FB
• If hitter guessing slider, pitcher can choose any
• Slider is dominated for pitcher he can always
  do at least as well choosing FB or CU
• Stepping through hitters choices given pitcher
  strategy (examining by rows)
• Slider is dominated for hitter he can always do
  at least as well will FB or CU
• ID condenses to upper (4) left cells
• Isolate these cells and look for
  dominant-dominated strategies
• No single (pure) strategy Nash Equilibrium
• Mixing strategies is best

17
Building Intuition

• Sometimes complexity of situation or lack of
  practice makes finding exact solution difficult
• Lessons for Being a Better Simultaneous ID
  decision maker?
• Does I or my opponent have a dominant strategy?
• Do I or my opponent have a dominated strategy

18
Mixing Strategies

• Pitcher-Hitter ID had no single best strategy
  whats the best way to mix FB and CU?
• Take relative gains-losses to each into account
  (See graph on next slide)

19
Best Mix in Pitching Example
20
Finding the Right Mix

• Optimal proportion fastballs about 65 --
  Intuition?
• Percentage declines when facing a better fastball
  hitter
• Percentage declines when facing a poor change-up
  hitter
• Percentage near 50 only when hitter about same
  against fastball and change-up
• Intuition from Poker mixing bluffs strong
  hands?
• No pure strategy -- mixing play of strong hands
  with bluffs is better than either strategy alone
• The optimal proportion bluffs is usually low
  (e.g. 10) because the risks are very high
• Bluffs should increase if players beliefs
  easily manipulated
• Randomly mixing in bluffs is required but harder
  than might appear

21
Learning to Be Unpredictable

• Unpredictability (randomizing) easy for a
  computer an art for humans
• MLB Pitcher Poker
• Lessons From Experiments
• Coin Matching
• 3-people -- no match pays reward
• Setup 1 Players know who rivals are and size of
  payoffs
• Setup 2 Players dont now rivals or size of
  payoffs
• Same setup except counterpart exact payoff
  unknown
• History of plays has effect
• Why?

22
of Heads Chosen Based on History of Play
Full Payoff Information
Incomplete Information
Heads Last Period
Heads Last Period
23
Lesson from Coin Matching?

• Setup 1 Players employ almost exact NE (5050)
  with randomized mix
• Past play history has no effect
• Setup 2 Players diverge from NE
• Past play history has effect more heads in
  past round means fewer heads now
• Why the difference?

24
Takeaways?

• Simultaneous or Sequential?
• Dividing a fixed pie or influencing pies size?
• Eliminate choices where decision maker can do
  better regardless of rivals strategy
• Look for a strategy that is always best
  regardless of rivals choice
• If multiple strategies are best
• Mix them based on relative gains/losses
• Randomize the appropriate mix
• Keep in mind the effects of information (or lack
  of it) on the way people solve problems