## Playing Games

• last time, we saw a game: Downsian model of competition
• assumptions: who are key actors + what they can do + what their goals are
• conclusion: prediction about what is going to happen
• game theory is a set of tools to get from assumptions to conclusions

## What is Game Theory?

• Roger Myerson
• game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers
• game Theory provides general mathematical techniques for analyzing situations in which two or more rational individuals make decisions that will influence one another's welfare
• rationality: actors maximize own welfare (utility)
• strategic interaction: actors' utility depends on what others do

## Elements of a Game

• players: $$N = \{1,\dots, n\}$$ ($$i \in N$$ is individual player)
• actions: player $$i$$ chooses action $$a_i \in A_i$$ ($$A_i$$ is set of her actions)
• profile of actions is $$(a_1, \dots, a_n)$$ specifies every player's action
• payoffs: utility of player $$i$$ is given by $$u_i(a_i,a_{-i})$$

## Elements of a Game: Advanced

• timing
• static: all players move at the same time
• dynamic: players move sequentially / in some order
• information
• complete: everything about the game is common knowledge
• incomplete or imperfect: some players uninformed about payoffs or identities of other players, possible strategies, outcomes, knowledge, etc

## Game Example: Fighting Countries

• Consider two countries $$A$$ and $$B$$ involved in a territorial dispute (e.g., over Antarctic territory). $$A$$ and $$B$$ must decide whether to fight. If both decide to fight, a war breaks out and the cost of war is greater than the value of the territory. We assume that in this case, both countries get a utility loss of $$-1$$. If only one country decides to fight, it gets the territory. The payoff of obtaining the territory is $$2$$ for country $$B$$ and $$3$$ for country $$A$$. A country which does not fight always get $$0$$. Suppose that both countries make their decision simultaneously.

## Nash Equilibrium: Definition

• Nash Equilibrium is a system of best responses

• each player maximizes utility given what other players do
• formal definition: consider a simultaneous game with $$N$$ players. Profile of actions $$(a_1^*,\dots, a_n^*)$$ is a NE if for every player $$i \in N$$,

$$u_i (a_i^*,a_{-i}^*) \geq u_i (a_i,a_{-i}^*), \text{ for every other action } a_i \ne a_i^*$$

## Solving Fighting Countries

• static games of complete information with finite sets of actions $$\to$$ matrix form
Country B
Fight Not Fight
Country A Fight $$-1,-1$$ $$3,0$$
Not Fight $$0,2$$ $$0,0$$
• Nash equilibrium (intersection of best responses):
• (Not Fight, Fight) and (Fight, Not Fight)

## Another Example: Political Factions

• two factions ($$L$$ and $$R$$) within the same party decide whether to support $$\text{left candidate}$$ or $$\text{right candidate}$$
• if both factions support the same candidate, the candidate wins primary and general election (factions receive payoffs of $$2$$)
• factions prefer different candidates:
• faction $$L$$ prefers $$\text{left candidate}$$ and receives $$+8$$ utility if $$L$$ wins general election
• faction $$R$$ prefers $$\text{right candidate}$$ and receives $$+8$$ utility if $$R$$ wins general election
• if factions support different candidate, general election is lost, and factions receive $$0$$

## Solving Factions Game

Faction $$R$$
$$\text{candidate L}$$ $$\text{candidate R}$$
Faction $$L$$ $$\text{candidate L}$$ $$10,2$$ $$0,0$$
$$\text{candidate R}$$ $$0,0$$ $$2,10$$
• Nash equilibrium: ($$\text{candidate L}$$,\ $$\text{candidate L}$$) and ($$\text{candidate R}$$,\ $$\text{candidate R}$$)

## Example: More than $$2$$ Actions

left middle right
top $$2,0$$ $$1,1$$ $$4,2$$
center $$3,4$$ $$1,2$$ $$2,3$$
bottom $$1,3$$ $$0,2$$ $$3,0$$
• Nash equilibrium: (center, left) and (top, right)

## Example: Stag Hunt

hunter $$B$$
stag rabbit
hunter $$A$$ stag $$2,2$$ $$0,1$$
rabbit $$1,0$$ $$1,1$$
• Nash equilibrium: (stag, stag) and (rabbit, rabbit)

## Example: Matching Pennies

player $$B$$
player $$A$$ heads $$-1,1$$ $$1,-1$$
tail $$1,-1$$ $$-1,1$$
• Nash equilibrium: cannot find one using the tools we have so far

## Nash Equilibrium: the Good and the Bad

• self-enforcing prediction

• if each player conjectures what others are playing...
• and each player is correct...
• then rationality predicts Nash equilibrium
• some issues:

• if multiple NE then unclear which one is better prediction
• no story behind HOW players get to NE

## Continuous Action Spaces

• interesting games usually have more than 2 players who have more than 2 actions
• if 2 players, more than 2 actions each $$\to$$ bigger matrix
• many situations are naturally continuous
• firms choosing price
• workers choosing hours
• politicians choosing policy platforms
• if some player's action space is continuous, we cannot represent it in matrix form
• solution is still NE, but solving technique is quite different

## Example: Public Goods Game

• players: person $$A$$ and person $$B$$, each has budget $$\20$$
• action: person $$i$$ chooses contribution $$x_i \in [0,20]$$ to the public good
• consumes the remaining balance of $$\20-x_i$$ privately
• payoff of person $$i$$ is

$$u_i(x_A,x_B) = (20-x_i) + 1.5 \times \left( \frac{x_A + x_B}{2} \right)$$

## Algorithm for Solving Static Games of Complete Information

• step 0: identify players, actions, payoffs
• step 1: identify best responses $$a_i^* (a_{-i})$$
• depending on everyone else's actions $$a_{-i}$$, what should $$i$$ do?
• finite/matrix games: fix a column, pick action with highest payoff for row player; fix row, pick best action for column player
• continuous actions: $$i$$'s best response is a function of $$a_{-i}$$
• step 2: NE is intersection of best responses