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