# The Prisoner's Dilemma and the Nash-Cournot Equilibrium

Figure 1 illustrates the loss function of each country define a set of indifference curves as ellipses in m1 and m2 space. Suppose each country wishes to minimize a loss function of the form: The indifferent curves have their particular shape because of the potential spillover from country 1 to country 2. Complete policy independence for the two countries in the diagram by vertical straight-line indifference curve for the home country and the horizontal straight line for the foreign country.

All the bliss points at which the indifference curves are tangential will represent the efficient, or optimal, policies for the two countries.

Using this Nash reaction function may be illustrated for country 1 using the geometric argument in the diagram. The Nash reaction function for country 1 defines the policies which generate minimum losses given the policy following by country 2. Figure 1. Construction of the Nash Reaction Function for Country 1 Figure 2 is a "Hamada diagram" which provides a general illustration of the case of two countries.

Both countries are assumed to have preferences regarding economic goals (output, employment, inflation, etc.

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). If other variables which are not dependent on policy actions are taken as given, these goals can be depicted solely as functions of the policy instrument values chosen by each country. For simplicity, each axis can be thought of as measuring just one policy instrument, though in the diagram as shown it is assumed that an index which combines these instruments can be used to scale each of the axes.

It is also assumed that a point like B1, and that of country 2 by a point like B2 gives the bliss point of country 1, in terms of these policy instrument values.

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These points correspond to the minimum possible losses for country 1 and country 2 respectively. Without cooperation, the world economy finds the equilibrium at point N where the Nash reaction functions intersect. The contract curve is shown on the diagram as joining the two bliss points B1 and B2. The diagram also shows the reaction curve for each country. This simply plots the policy instrument values of the one country responding to the other's choices.

It is constructed to represent the country in question setting its own policy instruments to achieve the best outcome for it, given the other country's policy stance. (Corden, 1985)Since achieving the best outcome here means reaching the highest possible preference curve, the reaction function for country 1 (NRF1) runs through all the points of tangency between vertical lines (which represent country 2's given policy values) and the indifferent curve for country 1. And country 2's reaction function (NRF2) joins the points of tangency between horizontal lines and country 2's indifferent curves.

Figure 2. Nash Equilibrium with Incompatible Exchange Rate Targets in the Two Country Model In the figure 3, there are three types of equilibrium are represented. N, the Nash equilibrium, is the point at which the two reaction curves intersect; S, the Stackelberg equilibrium, with country 1 as leader; and the contract line joining B1 and B2 being the locus of possible cooperative solutions. The country optimizing myopically forms each curve, which is taking as given the other country's policies. Hence the intersection depicts the position, which emerges when both countries behave in this way.

S is the Stackelberg, or leadership, with country 1 as the leader. Country 1 is not myopic. It determines its own policies so as to force country 2 to adopt policies, which, in combination with its own, provide it with the best attainable level of welfare. This means, S is on country 2's reaction curve, at the point most advantageous to country 1 (a point of tangency with one of country 1's preference curves). Notice that although country 1 is the leader, both country 1 and country 2 may be better off under its leadership than under the

myopic Nash regime, which means that S is on better indifferent curves for both countries than is N. Finally, there is the cooperative solution. This is the locus of points of tangency between the indifferent curves of the two countries. Thus the solution is Pareto-optimal: an improvement for both countries simultaneously is not possible from a point on the curve. Which point on this locus is actually chosen depends on a variety of factors, including 'bargaining strength'. Figure 3. Stackelberg Leader Solution in the Two Country Model with Incompatible Exchange Rate Targets

Coordinated policies are more desirable than uncoordinated policies, but unfortunately, policymakers generally have an incentive to cheat in these Pareto-improving outcomes, and politically sovereign policymakers seem to have difficulty achieving them. For example, the home country believes the other country will maintain its instrument at a coordinated equilibrium on the contract curve, it would be in the interests of the home country to move to its reaction function and so obtain a higher level of utility, in this instance the gains from coordination will not be reliable.

The cheating problem is compounded by the difficulty of defined and verifying cooperative problems, and by the moral hazard this implies. (Canzoneri and Gray, 1985) There are two types of cheating, one is domain cheating, another is player-domain cheating. The former type of cheating has a Stackelberg nature because of the lack of a retaliatory instrument after time zero, whilst the latter is a Nash-type problem in player space. Both types of cheating are problems covered by the general concept of reputation and, from a policy coordination perspective, player-domain cheating is likely to be the most important component of reputation.

(Hallwood and MacDonald, 2000) Optimal cooperative policies depend on the objectives of the policymakers, the nature of the transmission mechanism between the economies, the policy tools that they have available, and the nature of the disturbances that hit their economies and call for policy responses. Differences in objectives between countries affect the basic principle of gains from cooperation. The ongoing nature of policy interactions among countries, reputational considerations make cooperative equilibrium more likely. (Barro, 1986)Each country knows it will be better off in the long run if the cooperative equilibrium is maintained.

Countries may develop strategies both to punish those that do not cooperate, and to earn a reputation for reliability. It then becomes possible that countries will reach and stay at the cooperative equilibrium. Coordination through reputation, without explicit international agreements, is less likely the more countries there are. When everyone is at the cooperative equilibrium, the temptation for one small country to break ranks is very strong. The potential cost to it of doing so may also be high, for it is more dependent on the world economy than is a larger country.

But because it inflicts very little damage on the rest of the world by not cooperating, it is not certain that it will be penalized. Finally, in the absence of coordination, policies which rely on reputation may be undesirable. (Currie et. al. , 1987) This is because the government which has reputation can more readily affect market expectations, and the coordination failures, especially those relating to the exchange rate, that arise from the noncoordination of policies are more likely to be increased when a government has reputation.

From all above issues, which discussed on policy coordination, considering the game theory relating to the optimality of cooperative strategies, then, make a convincing case that coordination is generally superior to noncooperative policy-making.

## References

1. Barro, R. J& D. Gordon (1983), 'Rules, Discretion and Reputation in a Model of Monetary Policy', Journal of Monetary Economics, Vol. 12, pp. 101-121
2. Barro, R. J (1986), 'Recent Developments in the Theory of Rules Versus Discretion', Economic Journal, Vol. 96, pp. 23-27
Updated: May 19, 2021