I (Dr Maria Polukarov, University of Southampton) visited Dr Vangelis Markakis in Athens University of Economics and Business from 5/02/2011 to 18/02/2011. Within a broader topic of “Algorithmic and Conceptual Issues in Games Played in Structured Environments” of our collaborative work, during this visit we focused on Cournot oligopolies on graph structures.

Cournot games describe a fundamental model of competition between firms where they control their production levels and by doing so influence the market prices. In the simplest Cournot model all the firms produce the same good; the demand for this product is linear in the total production (i.e. the price decreases linearly with total production); the unit cost of production is fixed and equal across all firms.

We enrich this model with a social context graph with the meaning that each producer's quantity and price for the product is only affected by the actions taken by his neighbours in the graph. Unlike the traditional Cournot games that always possess an equilibrium, it is easy to construct examples of Cournot games on certain families of graphs where an equilibrium does not exist. During the visit, we managed to obtain the following negative results:

Theorem 1. There is no Nash equilibrium on a Cournot game when the graph

- contains a node that has more than $2$ neighbours of degree $1$ (note that a star with at least $3$ leaves is a special case).

- contains a node that connects two subgraphs in each of which each node has the same set of neighbours.

On the positive side, we were able to obtain the following results:

Theorem 2.

There exists a unique Nash equilibrium when the graph is

- the complete graph $K_n$

- a complete bipartite graph $K_
$ ($r+t = n$).

- a regular graph.

- a parallel graph.

- a line.

- an odd cycle. For even cycles, there is an infinite number of equilibria. Moreover, the unique equilibrium of odd cycles is a valid equilibrium for even cycles.

We plan to continue this work, with the aim of characterizing the graph structures that admit a Nash equilibrium and also generalize our study to more complicated models of oligopolies.

Cournot games describe a fundamental model of competition between firms where they control their production levels and by doing so influence the market prices. In the simplest Cournot model all the firms produce the same good; the demand for this product is linear in the total production (i.e. the price decreases linearly with total production); the unit cost of production is fixed and equal across all firms.

We enrich this model with a social context graph with the meaning that each producer's quantity and price for the product is only affected by the actions taken by his neighbours in the graph. Unlike the traditional Cournot games that always possess an equilibrium, it is easy to construct examples of Cournot games on certain families of graphs where an equilibrium does not exist. During the visit, we managed to obtain the following negative results:

Theorem 1. There is no Nash equilibrium on a Cournot game when the graph

- contains a node that has more than $2$ neighbours of degree $1$ (note that a star with at least $3$ leaves is a special case).

- contains a node that connects two subgraphs in each of which each node has the same set of neighbours.

On the positive side, we were able to obtain the following results:

Theorem 2.

There exists a unique Nash equilibrium when the graph is

- the complete graph $K_n$

- a complete bipartite graph $K_

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**r**n'a pas pu être exécuté.- a regular graph.

- a parallel graph.

- a line.

- an odd cycle. For even cycles, there is an infinite number of equilibria. Moreover, the unique equilibrium of odd cycles is a valid equilibrium for even cycles.

We plan to continue this work, with the aim of characterizing the graph structures that admit a Nash equilibrium and also generalize our study to more complicated models of oligopolies.