January 2025

The article at a glance: Repeated interactions among economic agents feature a variety of possible equilibrium outcomes, making it hard to obtain predictions. We introduce a protocol of renegotiation, which refines the set of equilibria to essentially a unique outcome, for a class of games between two agents. In that outcome, an agent gets payoff proportional to their benefit from deviating from cooperation. This prediction can be empirically tested. 

Strategic Renegotiation in Repeated Games

by Mikhail Safronov, CERF Fellow, Associate Professor, Faculty of Economics, University of Cambridge

Renegotiation in repeated interactions: good and bad.

Repeated games are widely used to model interactions between economic agents. One feature of repeated games is that there may be many possible equilibrium outcomes, especially if agents are patient enough. For example, an inefficient scheme in which agents interact and benefit every odd month, and do nothing every even month can be an equilibrium. In this paper Bruno Strulovici and I challenge this vast variety of outcomes by allowing agents to occasionally meet and negotiate, and mutually agree to change their behaviour if it benefits everyone. This possibility can help agents, and it can also create troubles. To make sure that nobody deviates from efficient interaction, one may need a threat of inefficient punishment. For example, if in a cartel a firm lowers its prices and benefits at the cost of other cartel members, other firms will also lower their prices in the future. Everyone will be hurt, including the deviator. The presence of negotiation makes such punishment schemes less effective, since agents may agree to return to the efficient cooperation too soon, effectively forgiving the deviator.

Good times: Efficiency. Bad times: The shortest.

In our model we impose two rules of renegotiation. Firstly, whenever agents get a chance to renegotiate, they always agree on efficient cooperation. Secondly, agents always aim to renegotiate as often as possible. Put it differently, we look at the shortest duration of punishment that still allows for efficient cooperation. This leads to a unique equilibrium outcome in a class of games. To demonstrate the result, consider Bertrand competition among firms. Suppose firms form a cartel, they agree to charge the monopoly price, and somehow split the monopoly profit. If any firm deviates and lowers it price, firms will engage in price war by charging low prices for a certain duration. Any firm will be hurt by not having their share of monopoly profit. Hence, to prevent deviation, each firm must get at least a certain minimal share of profit during cooperation. A shorter duration for price war increases these minimal shares. When the sum of these minimal shares becomes one, this would correspond to the shortest possible length of punishment, and the related minimal shares will yield an essentially unique equilibrium outcome. Such a way of splitting surplus may definitely cause questions. Why should agents always negotiate to move to an efficient outcome, since it effectively leads to forgiving the deviator? While there are other potential rules of renegotiation, the current scheme is attractive in that it yields a uniqueprediction for certain scenarios, which can be tested empirically.

Renegotiation may prevent productivity.

In the unique equilibrium outcome, every agent gets their payoff proportional to their benefits from deviation. Any agent may wish to have these benefits high, to have better positions in negotiation. In some scenarios, for example a cartel, a firm may wish to invest in a new technology and decrease its production costs, since it will be more profitable to deviate. In some other scenarios an agent may wish not to become more productive. If efficient cooperation requires each agent to make a costly contribution to a certain project, then having smaller costs may hurt the agent. In this case deviation would be not to contribute, and the benefit of deviation will be proportional to costs. Lowering these costs will lead to a lower payoff for the agent.

Proceed with caution.

For any theory, it is important to be aware of the circumstances when its results do not hold. For the current model, the result of having the unique outcome holds for certain games, where one needs an inefficient punishment to sustain cooperation. There exist cases when this is not true. In a Prisoners’ Dilemma one does not need inefficient punishment, hence renegotiation has no bite. There are other important assumptions, which may be of interest to those familiar with game theory. The result only applies if there are two agents. It is also essential that agents do not choose so-called mixed strategies during punishment. An example of mixed strategies is a football striker taking a penalty kick, who should not reveal where they will kick the ball. Yet we believe the model can be useful for analyzing the effects of renegotiation in repeated interactions.