Why worldwide collaboration can be up to 5 times more efficient than selfish behavior
Using game theory to understand the price of selfishness in times of pandemic and climate change
Have you ever wondered why we need all these international organizations that try to make all 195 countries of the world work together towards a common goal? You’ve probably heard from other people (or even presidents of certain countries) that it is just a waste of time and money, right?! Wouldn’t it be much easier if we just let everybody clean their mess?!
Surprisingly, the consequences of opting for such an approach can be catastrophic on a global scale, and game theory provides us with a clear explanation of why this is the case.
Social optimum and selfish behavior
To understand the consequences of selfish behavior, I will use the example of the ongoing COVID-19 pandemic where the different actors (also called agents) are represented by independent countries,
and the cost they are paying when facing the pandemic is quantified by the expenses (humanitarian or economic) required to contain it. To analyze the effect of collaboration in this gloom context, let‘s consider the following scenario:
- We assume that there is one country (for instance, that from which the virus originates) that pays the heaviest tribute. Let’s denote its overall loss by 1 (it can be 100 or 100K, 1 is used for the sake of simplicity).
- Each country to which the virus spreads afterward can deal with it more efficiently when compared to the previous one. Let’s say that the second country can contain the pandemic by paying only half of what the first one has paid (1/2), while the third pays only one third (1/3), and so on. The final country number k will pay only 1/k fraction of the most affected country.
- We assume that the socially optimal outcome, however, would have been to contain the pandemic in the first country by asking every existing country to participate equally in the expenses needed for that. Let’s say that this optimal outcome has a total cost of (1+a) where a > 0 is a tiny overhead that accounts for the cost of mobilizing the required resources for the first suffering country.
Note that while the first two assumptions do not reflect how everything happened in reality, the current situation can still be reduced to this model by sorting the countries based on their losses due to the pandemic in descending order.
It is straightforward to see that in the case of our model, the overall cost of all countries deciding not to collaborate is equal to the harmonic number
that behaves as shown in the plot below:
The difference between the selfish cost and the social optimum (denoted as collaborative baseline) is pretty huge, right?! You may wonder why anybody would opt for such an additional loss when having a much more efficient option at hand.
Well, that is where game theory comes into play with the concept of Nash equilibrium, and to explain it, let us consider the different choices available to the agents in our game.
Let us start with the last country from the example given above. In our model, this country is offered a choice to pay 1/k to contain the pandemic on its own or to pay (1+a)/k alongside other potential collaborators to achieve the overall socially optimal cost of 1+a.
When being selfish, this country will choose the option of paying 1/k to its benefit. Once this has happened, the country (k-1) faces a choice of paying (1+a)/(k-1) with others or opting for 1/(k-1) when dealing with the situation on its own.
Once again, it chooses selfish behavior for the same reason as the country before. In the end, the selfish approach leads to the notion of Nash equilibrium: every country adapts the strategy that minimizes its costs, and no country can do better by unilaterally changing their strategy. The overall cost at the Nash equilibrium is then the harmonic number defined above.
Price of stability
The ratio between the cost at Nash equilibrium and the social optimum is commonly referred to as the Price of Stability (PoS).
Price of Stability quantifies the potential loss of efficiency between the best outcome of the selfish behavior (best Nash equilibrium) and the social optimum of a given game involving a set of strategic agents.
In our case (the case of the so-called fair cost-sharing games studied by Anshelevich et al., FOCS’04), the PoS is bounded by the harmonic number defined above with k equal to 195 existing independent states in the world. If our very rough model of the pandemic is close to the truth, then the lack of worldwide collaboration can come at an alarming cost of
Once again, if our model, despite its over-simplistic setting, is somewhere near to be correct then the lack of collaboration can cost up to 5 times more than the social optimum. Also note that this goes beyond my example with the COVID pandemic and extends to any situation where independent states have a choice between solving the problem on their own or opting for a world-wide collaboration.
For instance, it highlights the importance of the Paris agreement signed by 195 countries to engage in the fight against climate change together and means, that we may well be avoiding the scenario of a drastic loss of efficiency with a possible devastating effect on our future.
Fair cost sharing games of our life
While my article may look like a simple illustration of an abstract game-theoretical result when applied to a situation of high-societal importance, its message is more general and destined for every one of us and our everyday choices.
Indeed, despite some seeming differences, all people on planet Earth play a game similar to that described above, albeit sometimes without even knowing it. Should I take the bus and share the travel cost with other passengers, or should I stick to using my car? Should I opt for the sharing economy platforms,
or should I personally possess all the goods that I need? All these are fair cost-sharing games of our everyday life, and all of them are subject to the inefficiency that becomes more pronounced when more people are concerned with it.
To push this idea even further, it won’t be an exaggeration to say that we are often tempted to avoid individual sacrifices (the 1/k or (1+a)/k situation) for an immediate gain and thinking that one person’s action will not change much.
But as my example shows, one person’s action can be enough to launch an unprecedented change to the behavior of the others. Indeed, if only the very first country were to choose to sacrifice its infinitesimal gain, it could have launched an incentive for others and make the collaborative alternative more attractive to them.
I agree that worldwide collaboration may not be easy at all. But the first step is to become conscious about its benefits. The next step is to think about it when you face any such choice in your life. And who knows maybe the step after that would be to enjoy a better world to live in.