Millennium Post

Taking the right call

Application of Game Theory to analyse public goods dilemma in the international and domestic arenas

Continuing with the theme of public goods, it would be instructive to analyse these issues from a Game Theoretic perspective. Last month, in these columns, I had discussed the grave public policy challenges facing us in the areas of Water, Air and Forest conservation. These are public goods and since they are non-excludable and subtractable, they tend to be overconsumed with underinvestment in their management and conservation, leading to the tragedy of the commons. I had suggested the tools of social capital and institutions to overcome these dilemmas.

I. Game Theory as a Framework

Recall that the right institutions and 'rules-in-use' can be a foil to the over consumption of public goods. However, the problem with such rules-in-use are many: who will make the rules, who will supervise and monitor such rules, how to overcome the problem of asymmetric information etc. In other words, the public goods dilemma can be reduced to issues of supervision and monitoring. Such dilemmas can also be analysed from a Game theoretic perspective.

As we know, game theory is a branch of mathematics and involves an analysis of strategies and decisions by various actors. Game Theory has been widely used in analysing public policy questions and modelling real life situations for a better understanding. Typically, in a simple model, there are two players who have two choices (called strategies). Each strategy leads to an outcome for the player called 'payoffs'. Various types of games can be modelled but for the purposes of this article we will discuss three games: (a) the Prisoners' Dilemma, (b) Chicken Game and (c) Assurance Game.

In general, the payoffs to players in any game look like the following:

Table 1

Player B C D

Player A

C (b, b) (d, d)

D (d, c) (e, e)

Where C and D are strategies, denoting 'cooperate' and 'defect' respectively and b, c, d and e are payoffs associated with these strategies.

(a) Prisoners' Dilemma Game: In this game, two robbers (players) are questioned separately by a Police Inspector. The two players have two strategies: co-operate (C) with the Police and confess or defect (D) from the Police by remaining silent. They however can't communicate with each other. The game is presented in Table 2.

Table 2

Robber B


Robber A

C (2, 2) (0, 3)

D (3, 0) (1, 1)

The above table shows the payoffs or outcomes of various possibilities. If the robbers confess, they get 2 years of jail (C, C) but if they remain silent or defect (D, D) they get 1 year of jail. It is logical to see that both prisoners, in their quest to minimise their punishment (with zero years in prison) would betray the other and end up with a sub-optimal outcome by confessing (2 years each in jail). This is clearly worse than spending one year each in jail, which would have been the outcome had they chosen to lie to the Police (D, D). The sub-optimal outcome where both confess is referred to as a Nash equilibrium named after John Nash, the winner of the Nobel Prize in Economics in 1994. The dilemma is that whatever the other robber does, each is better off confessing rather than remaining silent (referred to as a dominant strategy). In other words a sub-optimal outcome results.

Hence d > b > e> c in any Prisoners' Dilemma game.

(b) Chicken Game: This is a conflict game and involves two players in a headlong collision. If one player swerves, he is called 'chicken' and the other wins. In this game, the structure of payoff changes as compared to the Prisoners' Dilemma Game. The payoffs are now: d > b > c > e. Here the 'Suckers' payoff or the payoff for co-operating (C) when the other player defects is greater than the payoff that both player get from defecting. A typical payoff matrix is a Chicken game looks like the following:

Player A

Player B


C (8,8) (6, 10)

D (10, 6) (2, 2)

In a Chicken Game, there are two Nash equilibria (underlined). However, which equilibrium prevails will depend on the expectation of each player. For example (10, 6) will be the equilibrium if Player A wants the good badly enough.

(c) The Assurance Game:

In this game the payoffs are tweaked and now b > d > c > e.

In other words, the incentive structure is altered in such a manner that defecting singly or collectively is costlier than co-operating.

Illustrating with numbers, an Assurance Game looks like this

Player 1

Player 2


C (8,8) (1, 6)

D (6, 1) (2, 2)

Even though there are two equilibria in Assurance Games, the Pareto-optimal one is where both players (say developed and developing countries) co-operate.

II. Application to Public Goods Dilemmas

Recall that I had discussed the public goods dilemma in two broad areas: (A) In the international arena, the dilemma is playing out in world trade, financial crisis and climate change. (B) In the domestic arena, these dilemmas were analysed in Water, Air and Forest conservation efforts.

Let us model these dilemmas in a game theoretic framework.

(A) International Arena:

We will illustrate the climate change crisis using Game Theory. There are two players: Developed Countries and Developing Countries. The options before the countries are either to cut emissions and thereby cooperate or continue emissions and defect from the agreement.

Developing Countries

Developed Countries Cut Emissions No Cut

Cut Emissions (b, b) (c, d)

No Cut (d, c) (e, e)

There is an external monitor, say, the UNFCC (United National Framework for Climate Change) which is mandated to coordinate the actions of various countries in meeting emission targets (or the benefits accruing). The payoffs are as follows:-

i) Here e is the payoff when both players don't cut emissions, which will give a large benefit in the form of accelerated growth albeit at a high environmental cost.

ii) d is the payoff for not cutting emissions, when the other player has cut emissions

iii) b is the payoff when both developing and developed countries agree to cut emissions. This payoff is obviously less than what a country would get if it continues emissions since green technology is expensive and the high cost of transition from a coal-based economy to a 'renewable' economy would take its toll on growth.

iv) Finally, c is the payoff that accrues if one member cuts emissions but the other doesn't do so. In popular game theory parlance, this is also called the 'sucker's payoff' since the net result is continued emission by the defaulter and no progress towards the emissions objectives set by UNFCC.

Typically, what we are witnessing is a situation where d > b > e> c, and the incentive is loaded against cutting emissions. This is a classic Prisoners' Dilemma situation which is so often witnessed in the case of the supply of public goods. There is continued resort to polluting ways to maintain a high growth rate and a certain standard of living, by developed countries. Seeing such a high carbon footprint per capita of most developed countries, there is little incentive for developing countries to cut emissions. Even though countries such as India and China have set voluntary roadmaps to cut emissions, the real impact on global warming will come only if the US comes on board. To add to this dilemma, the new and 'green' technologies are expensive and developed countries don't easily part with them.

Hence, the challenge before the developing and developed countries is to recognise the principle of common but differentiated responsibility which India has been arguing at various world forums such as the WTO and UNFCC. This will also help convert the Prisoners' Dilemma game, in which the current climate change negotiations are struck into an Assurance Game (d > b> c > e). It is therefore not a surprise that the recent COP 25 meetings in Madrid on climate change collapsed without any conclusion.

(B) Domestic Arena:-

In the domestic arena we had analysed the issue of air pollution, water pollution and deforestation and seen that the tragedy of the Commons was unfolding because of lack of supply of public goods. We will use Game Theory to analyse such a dilemma in one area viz., deforestation.

As we have done above in the international example, we assume that there are two players in the forest areas. Cooperation between them requires regular patrolling of the forests and adhering to rules set by the community or a Forest Committee and defecting means that they violate the rules laid down by the Forest Committee. The payoff from cooperation and defection are listed as under:

i) Cooperation between two players means regeneration of the forests. So payoff is 'b' for both players.

ii) Defection by both players means that they over exploit the forest and ignore the rules leading to large scale deforestation. The payoff for this is zero.

iii) After one of the players cooperates and other defects, the latter gets benefits such as firewood and over grazing while the former who has adhered to the rules gets nothing or the 'suckers' payoff. Sucker's payoff is 'c' and the defector's payoff is 'd'.

Again, as discussed in the international example, if d > b > e >c, we are stuck with a prisoners' dilemma game which is a kin to the public goods dilemma where there is an over consumption of the public goods (forest in this case) and under supply of conservation of forests.

The effort of the local governments and the Forest Committee should therefore be to convert this prisoners' dilemma to an Assurance game as discussed earlier. For this, the payoff for defecting or free riding (d) must be smaller than the payoff obtained from cooperation (b) unlike in a prisoners' dilemma where 'd' is greater than 'b'.

III. Conclusion

We have argued that institutions need to change incentive structures so that public goods dilemma get converted from a prisoners' dilemma situation to an assurance game situation. Hence, in the international arena all countries need to discipline themselves on cutting emissions so that targets set for global warming and other parameters are within reach in the near future. Similarly, in the domestic arena Members of Forest Committees, RWAs and other Community groups need to discipline themselves in the use of common pool resources so that there is cleaner air, purer water and denser forest to enjoy for the future generations. The Assurance game provides a higher probability of solving the public goods dilemma or the free rider problem but it requires a lot of hard work in terms of funds and communication, strong leadership and sacrifice. The Assurance Game is also called a Cooperation Game for this reason. Let us all put our heads together to achieve this desirable objective in both domestic and international arenas.

Dr. Krishna Gupta is the Principal Resident Commissioner, Government of West Bengal. Views expressed are strictly personal

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