Econ 430 Lecture 9: Games on Networks

Transcription

1 Alper Duman Izmir University Economics, May 10, 2013

2 Semi-Anonymous Graphical Games refer to games on networks in which agents observe what their neighbours do The number of agents taking a specific action matters; not the identities. Formally, a semi-anonymous game is a graphical game such that the payoff to a player i with a degree d i who chooses action x i is described by a function u di (x i, m) where m is the number of players in N i (g) taking action 1.

3 An agent taking action 1 contributes to a local public good. A player who has m neighbours take action 1 gets a payoff u di (x i, m) = f (x i + λm) c i where f is nondecreasing function, and λ > 0 and c > 0 are scalars.

4 A Couples Game ; playing tennis A player prefers to take action 1 if at least one neighbour takes action 1, otherwise prefers action 0. Thus u di (1, m) = 1 c if m 1 u di (1, 0) = c u di (0, m) = 0

5 A semi-anonymous graphical game exhibits strategic complements if it satisfies the property of increasing differences; that is for all d and m m u di (1, m) u di (0, m) u di (1, m ) u di (0, m ) A semi-anonymous graphical game exhibits strategic complements if it satisfies the property of increasing differences; that is for all d and m m u di (1, m) u di (0, m) u di (1, m ) u di (0, m )

6 The best-shot public goods game is one of strategic substitutes. If f is concave local public goods game is also one of strategic substitutes. Why? The couples game is one of strategic complements. Investing in higher education by observing neighbours.

7 Semi-anonymous graphical games with strategic complementarities always have a pure strategy equilibria. Semi-anonymous graphical games with strategic substitutes do not always have a pure strategy equilibrium, but they always have at least one equilibrium mixed strategies. Which one is more likely?

8 Bramoulle and Kranton -Local Public Goods game Instead of a discrete set of actions, the action x i is continuous. Players benefit from their own actions plus the actions of their neighbours with payoff described by u i (x i, x Ni (g)) = f (x i + j N i (g) x j) cx i where f is a continuously differentiable, strictly concave function and c > 0 is a cost parameter.

9 Interesting case occurs when f (0) > c > f (x) for some x; otherwise players contribute 0 or infinite. Let x be such that f (x ) = c. Any pure strategy eq. must have at least x produced in each player s neighbourhood. There are subsets called specialized equilibria in which some agents provide x and the others free-ride.

10 All equilibria in this public goods game are inefficient. They do not maximize total utility. Think of a dyad, the total production is x. which maximizes f (x) cx Overall societal utility is 2f (x) cx which has higher maximizer as f is concave.

11 Ballaster et. al. Quadratic Payoffs and Strategic Complementarities Let x i is the intensity or effort so that higher x i corresponds to greater action. A player i s payoff is described by u(x i, x i ) = a i x i b i 2 x i 2 + j i w ijx i x j where a i 0 and b i 0 are scalars and w ij is the weight that player i places on j s action.

12 If w ij > 0 then i and j s activities are strategic complements. The expression b i 2 x i 2 leads to diminishing returns to activity Take the derivative of u(x i, x i ) and equate to 0. Solution is x i = a i + w ij j i x j b i b i

13 Let g ij = w ij /b j and set g ii = 0. Then In vector terms x = α + gx where x is the nx1 vector of x i and α is the nx1 vector of a i /b i If a i = 0 for each i then g = gx, so that x is a unit right-hand eigenvector of g. Otherwise, x = (I g) 1 α

14 By substitution we also get x = k=0 g k α Both expressions have much to the centrality measures we have discussed. If we let a i = a and b i = b for all i, we obtain x = (I 1 b w) 1 a b I This looks familiar to Katz-prestige and Bonacich centralities Indeed we can write x = a b (I + PKatz (w, 1 b ))