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 decisionmakers
 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 \(B\)  

heads  tails  
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

selfenforcing 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 \(\$20x_i\) privately
 payoff of person \(i\) is
\(u_i(x_A,x_B) = (20x_i) + 1.5 \times \left( \frac{x_A + x_B}{2} \right)\)
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