Games with Simultaneous Moves Nash equilibrium and normal form games

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

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

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.

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.

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

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.

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

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

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

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

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

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.

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.

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

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

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

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

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.

Caveats Maximin analysis only works for zero or constant sum games

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.

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

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

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

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

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

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

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)

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

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

Graphical Solution Price Demand MR Quantity

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

Graphical Solution – Part 2 Price Demand Pm MR Qm Quantity

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

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.

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

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

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)

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

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 = 6 - .5q2 (This is the best response function)

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 = 6 - .5q2 q2 = 6 - .5q1 Solving: q1 = q2 = 4

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 = 6 - .5q1 Knowing this, what should 1 choose?

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 = 6 - .5q1 Therefore, firm 1 will choose q1 to maximize Profit1 = q1 x (12 – q1 –(6 - .5q1)) Profit1 = q1 x (6 – .5q1)

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

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

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.

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

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

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

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

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

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

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.