1. Games with Simultaneous Moves
Nash equilibrium and normal form games

2. Overview
In many situations, you will have to determine your strategy without knowledge of what your rival is doing at the same time
Product design
Pricing and marketing some new product
Mergers and acquisitions competition
Voting and politics
Even if the moves are not literally taking place at the same moment, if your move is in ignorance of your rival’s, the game is a simultaneous game

3. Two classes of Simultaneous Games
Constant sum
Pure allocation of fixed surplus
Variable Sum
Surplus is variable as is its allocation

4. Constant sum games
Suppose that the “pie” is of fixed size and your strategy determines only the portion you will receive.
These games are constant sum games
Can always normalize the payoffs to sum to zero
Purely distributive bargaining and negotiation situations are classic examples
Example: Suppose that you are competing with a rival purely for market share.

5. Variable Sum Games
In many situations, the size and the distribution of the pie are affected by strategies
These games are called variable sum
Bargaining situations with both an integrative and distributive component are examples of variable sum games
Example: Suppose that you are in a negotiation with another party over the allocation of resources. Each of you makes demands regarding the size of the pie.
In the event that the demands exceed the total pie, there is an impasse, which is costly.

6. Nash Demand Game
This bargaining game is called the Nash demand game.

7. Constructing a Game Table
In simultaneous move games, it is sometimes useful to construct a game table instead of a game tree.
Each row (column) of the table corresponds to one of the strategies
The cells of the table depict the payoffs for the row and column player respectively.

8. Game Table – Constant Sum Game
Consider the market share game described earlier.
Firms choose marketing strategies for the coming campaign
Row firm can choose from among:
Standard, medium risk, paradigm shift
Column can choose among:
Defend against standard, defend against medium, defend against paradigm shift

9. Game Table – Payoffs Defend Standard Defend Medium Defend Paradigm
20%
50%
80%
Medium Risk
60%
56%
70%
Paradigm Shift
90%
40%
10%

10. Game Table – Variable Sum Game
Consider the negotiation game described earlier
Row chooses between demanding small, medium, and large shares
As does column

11. Game Table – Payoffs Low Medium High 25, 25 25, 50 25, 75 50, 25
50, 50
0, 0
75, 25

12. Solving Game Tables
To “solve” a game table, we will use the notion of Nash equilibrium.

13. Solving Game Tables Terminology
Row’s strategy A is a best response to column’s strategy B if there is no strategy for row that leads to higher payoffs when column employs B.
A Nash equilibrium is a pair of strategies that are best responses to one another.

14. Finding Nash Equilibrium – Minimax method
In a constant sum game, a simple way to find a Nash equilibrium is as follows:
Assume that your rival can perfectly forecast your strategy and seeks to minimize your payoff
Given this, choose the strategy where the minimum payoff is highest.
That is, maximize the amount of the minimum payoff
This is called a maximin strategy.

15. Constant Sum Game – Finding Equilibrium
Defend Standard
Defend Medium
Defend Paradigm
Min
Standard
20%
50%
80%
Medium Risk
60%
56%
70%
Paradigm Shift
90%
40%
10%
Max

16. Constant Sum Game – Row’s Best Strategy
Defend Standard
Defend Medium
Defend Paradigm
Min
Standard
20%
50%
80%
Medium Risk
60%
56%
70%
Paradigm Shift
90%
40%
10%
Max

17. Constant Sum Game – Column’s Best Strategy
Defend Standard
Defend Medium
Defend Paradigm
Min
Standard
20%
50%
80%
Medium Risk
60%
56%
70%
Paradigm Shift
90%
40%
10%
Max

18. Constant Sum Game – Equilibrium
Defend Standard
Defend Medium
Defend Paradigm
Min
Standard
20%
50%
80%
Medium Risk
60%
56%
70%
Paradigm Shift
90%
40%
10%
Max

19. Comments
Using minimax (and maximin for column) we conclude that medium/defend medium is the equilibrium.
Notice that when column defends the medium strategy, row can do no better than to play medium
When row plays medium, column can do no better than to defend against it.
The strategies form mutual best responses
Hence, we have found an equilibrium.

20. Caveats
Maximin analysis only works for zero or constant sum games

21. Finding an Equilibrium – Cell-by-Cell Inspection
This is a low-tech method, but will work for all games.
Method:
Check each cell in the matrix to see if either side has a profitable deviation.
A profitable deviation is where by changing his strategy (leaving the rival’s choice fixed) a player can improve his or her payoffs.
If not, the cell is a best response.
Look for all pairs of best responses.
This method finds all equilibria for a given game table
But it’s time consuming for more complicated games.

22. Game Table – Row Analysis
Low
Medium
High
25, 25
25, 50
25, 75
50, 25
50, 50
0, 0
75, 25
For row: High is a best response to Low

23. Game Table – Row’s Best Responses
Low
Medium
High
25, 25
25, 50
25, 75
50, 25
50, 50
0, 0
75, 25

24. Game Table – Column Analysis
Low
Medium
High
25, 25
25, 50
25, 75
50, 25
50, 50
0, 0
75, 25
For column: High is a best response to Low

25. Game Table – Column’s Best Responses
Low
Medium
High
25, 25
25, 50
25, 75
50, 25
50, 50
0, 0
75, 25

26. Game Table – Equilibrium
Low
Medium
High
25, 25
25, 50
25, 75
50, 25
50, 50
0, 0
75, 25

27. Summary In this game, there are three pairs of mutual best responses
The parties coordinate on an allocation of the pie without excess demands
But any allocation is an equilibrium

28. Strategies that are continuous variables
In many situations, it makes sense to model the strategies as being continuous rather than coming from a small set of choices
Classic example: competition in quantities (Cournot competition)

29. Cournot Competition
Suppose that two firms are competing by choosing quantities of goods to place on the market
Both have identical, constant marginal costs
Normalize these to be zero
Everyone faces a linear demand curve

30. Monopoly
If firm 1 anticipates that firm 2 will choose to produce NO output, then firm 1 is a monopolist in this market
We can find firm 1’s best response to this conjecture via the usual solution to monopoly problems

31. Graphical Solution
Price
Demand MR
Quantity

32. Comments Notice that marginal revenue lies below demand curve
Discounts to attract the next customer have to be passed along to all the other existing customers
Therefore, the increment to revenue is less than the willingness to pay of the marginal customer

33. Graphical Solution – Part 2
Price
Demand Pm MR Qm
Quantity

34. Other Conjectures
Now suppose that firm 1 conjectures that firm 2 will produce 10 units.
Firm 1 faces the following demand curve P Q 10

35. Comments
Notice that this just shifts the location of the y-axis in the standard diagram
The “best response” of firm 1 is calculated the same way---solve the monopoly problem for the “adjusted” demand curve.

36. Graphical Solution – Firm 2 produces 10
Price
Demand
MR1 10
Quantity

37. Graphical Solution – Optimal Firm 1 Production
Price
Demand P*
MR1 Q* 10
Quantity
Firm 1’s quantity

38. Some Math
We can write the graphical situation down as an optimization problem.
Firm 1 conjectures firm 2’s quantity, q2, and chooses its own quantity, q1, to maximize profits
Suppose demand is P = 12 – Q, then
Choose q1 to maximize
Profit1 = q1 x P
Profit1 = q1 x (12 – q1 – q2)

39. More Geometric Intuition
Profit1 = q1 x (12 – q1 – q2)
Notice that if q1 = 0, firm 1 earns no profits
Likewise if q1 = 12 – q2
Thus, the profit function looks like
Slope is flat at
The top
Profit q1

40. Calculus Profit1 = q1 x (12 – q1 – q2)
Differentiate with respect to q1 and find the q1 at the top of the hill (i.e. slope = 0)
This yields:
d Profit1/dq1 = 12 – 2q1 – q2
(This is the slope of the hill)
12 – 2q1 – q2 = 0
(This is the top of the hill – slope = 0)
q1 = q2
(This is the best response function)

41. Equilibrium An equilibrium is a pair of mutual best responses.
This means that each side conjectures the other’s move correctly and best responds to it.
So we need q1 and q2 solving
q1 = q2
q2 = q1
Solving: q1 = q2 = 4

42. Sequential Competition
Now let’s compare this to the situation where firm 1 moves first --- and is observed --- followed by firm 2.
Firm 2’s problem is just the same as it was before.
Therefore, firm 1 anticipates that if it chooses q1, firm 2 will choose
q2 = q1
Knowing this, what should 1 choose?

43. Firm 1’s Optimization
Firm 1 should look forward and reason back in making its decision
Recall:
Profit1 = q1 x (12 – q1 – q2)
But firm 1 knows (looking forward) that
q2 = q1
Therefore, firm 1 will choose q1 to maximize
Profit1 = q1 x (12 – q1 –(6 - .5q1))
Profit1 = q1 x (6 – .5q1)

44. Calculus
Once again, firm 1 seeks to get to the top of its profit hill:
Profit1 = q1 x (6 – .5q1)
dProfit1/dq1 = 6 – q1
The slope is flat at the top
6 – q1 = 0
q1 = 6
And, knowing q1, q2 = 3

45. Comments
Going first in this game (Stackelberg competition) enables firm 1 to gain market share at firm 2’s expense
Even though the market price is now lower…
(8 units of the good on the market when moves were simultaneous versus 9 now)
Firm 1’s profits are higher
How do we know this without calculating it?
This game has a first-mover advantage

46. How did firm 1 gain an advantage?
Commitment
Firm 1 could commit to produce more when its production decision was observable by 2
Strategic substitutes
Further, firm 1’s good is a (perfect) substitute for 2’s good.
By committing to produce more, firm 2 was obliged to scale back production
Thus, the two goods are strategic substitutes.

47. Other Archetypal Strategic Situations
We close this unit by briefly studying some other common strategic situations

48. Hawk-Dove
In this situation, the players can either choose aggressive (hawk) or accommodating strategies
From each players perspective, preferences can be ordered from best to worst:
Hawk – Dove
Dove – Dove
Dove – Hawk
Hawk – Hawk
The argument here is that two aggressive players wipe out all surplus

49. Hawk-Dove Analysis We can draw the game table as: Best Responses:
Reply Dove to Hawk
Reply Hawk to Dove
Equilibrium
There are two equilibria
Hawk-Dove
Dove-Hawk
Hawk
Dove
0, 0
4, 1
1, 4
2, 2

50. Battle of the Sexes
In this game, surplus is obtained only if we agree to an action
However, the players differ in their opinions about the preferred action
All surplus is lost if no agreement is reached
There are two strategies: Value or Cost

51. Payoffs
Suppose that the column player prefers the cost strategy and row prefers the value strategy
Preference ordering for Row:
Value-Value
Cost-Cost
Anything else
Preference ordering for Column

52. BoS Analysis We can draw the game table as: Best Responses:
Reply Value to Value
Reply Cost to Cost
Equilibrium
There are two equilibria
Value-Value
Cost-Cost
Value
Cost
2, 1
0, 0
1, 2

53. Conclusions This is called Nash equilibrium
Simultaneous games are those where your opponent’s strategy choice is unknown at the time you choose a strategy
To solve a simultaneous game, we look for mutual best responses
This is called Nash equilibrium
Drawing a game table is a useful way to analyze these types of situations
When there are many strategies, using best-response analysis can help to determine proper strategy
Games may have several equilibria.
Focal points and framing effects to steer the negotiation to the preferred equilibrium.