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 \(B\)
		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

• 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