Doklam standoff: Mathematician decodes whether China and India will go to war

To construct a Game Theory model, let us assume that the two players have to choose between three strategies.

 |  6-minute read |   11-08-2017
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It has been two months and both sides have refused to budge.  The Chinese insist on a unilateral Indian troop withdrawal while the Indians are adamant on mutual troop withdrawal. As the war clouds over Doklam refuse to recede, let us try and understand the standoff from the point of view of Game Theory.

Game Theory is a branch of mathematics that analyses how decisions are made in conflict situations. The decision-makers are the “players” in a “game”. Each player chooses from a set of strategies available to him. 

The choice of strategies by the players results in an outcome. Each outcome is associated with a payoff (expected utility) for the player. Players are assumed to be rational in that they would prefer a higher payoff to a lower one. And each player assumes that other players are rational in the same sense.

To construct a Game Theory model of the Doklam standoff, let us assume that the two players, China and India, have to choose between three strategies: Withdraw, Stay or Escalate. 

The Stay strategy means to maintain current troop levels and the Escalate strategy means to increase troop levels. There is a cost to each player if he chooses to Stay and a higher cost of mobilising troops if he chooses to Escalate. Since Escalation is only limited to increasing troop levels, it does not automatically result in war. 

Each player can then be reasonably assumed to have the following order of preferences for the outcomes:

Withdraw when other Stays (total defeat) < Withdraw when other Escalates (defeat) < Stay when other Escalates (partial defeat) < Both Escalate < Both Stay < Both Withdraw < Escalate when other Stays (partial victory) < Escalate when other player Withdraws (victory) < Stay when other Withdraws (total victory).

We can assign payoffs 0, 1, 2, 3, 4, 5, 6, 7, 8 for the above outcomes respectively, where the numerical values of the payoffs have no meaning except to represent the ranking of the preference.

We construct a payoff matrix whose rows represent the choices for India (row player) and the columns represent the choices for China (column player). The nine cells of the matrix then represent the nine possible outcomes.  

The payoffs are written in each cell as an ordered pair of numbers. The first number in the pair is the payoff for India and the second number is the payoff for China.

      China  
    Withdraw Stay Escalate
  Withdraw 5, 5 0, 8 1, 7
India Stay 8, 0 4, 4 2, 6
  Escalate 7, 1 6, 2 3, 3

It is clear from the payoff matrix that a player is always better off choosing to Stay rather than Withdraw, irrespective of the other player’s choice, i.e. the strategy to Stay dominates the strategy to Withdraw. Hence we can remove the strategy to Withdraw from both players’ choice of strategies.

This reduces the game to the following payoff matrix:

    China  
    Stay Escalate
India Stay 4, 4 2, 6
  Escalate 6, 2 3, 3

This is the classic game of Prisoner’s Dilemma. The dominant strategy in this game is to Escalate. This leads to the only equilibrium in the game, which is (Escalate, Escalate).

Note that the both India and China are better off if they end up at the (Withdraw, Withdraw) outcome, which is the Pareto Optimal outcome. This outcome is possible only if both sides cooperate.

Will the (Escalate, Escalate) outcome lead to war?

Let us consider that the players do not cooperate in the previous game.  This leads to the outcome where both sides have mobilised troops and are at a standoff.

The choices of strategy are to Stay or Attack. The Stay strategy means to stay in the Escalated position of the last game. Each player is assumed to have the following order of preferences for the four possible outcomes:

Both players attack (war) < Stay and other attacks (defeat) < Both Stay < Attack and other Stays (victory).

We can assign payoffs of 0, 1, 2 and 3 respectively for the four outcomes. Again, the numerical values of the payoffs have no meaning except to represent the ranking of the preference.

As in the last game we construct a payoff matrix whose cells represent the four outcomes.

    China  
    Stay Attack
India Stay 2, 2 1, 3
  Attack 3, 1 0, 0

This is the game of Chicken - A standard game used to describe a conflict situation in which two players appear to be heading for a clash.

In the Hollywood version of this game, two drivers race their cars towards each other. Each player has the choice of swerving or not swerving. The player who chickens out and swerves first gets a lower payoff than the player who holds his nerve and does not swerve. If neither player swerves then the cars collide and result in the lowest payoff for both players.

So what is the likely outcome of the standoff according to this model? It can be easily verified that here is no dominant strategy in this game. Hence there is no dominant strategy equilibrium. 

Let us consider the situation from the viewpoint of any one particular player. If this player is convinced the other player is going to Stay, then he is likely to attack rather than Stay. So the (Stay, Stay) outcome is not a stable outcome.

Similarly, if the player is convinced that the other side is going to Attack then he would prefer to Stay rather than risk a war. So the (Attack, Attack) outcome is also unstable. 

Since both players want to do the opposite of what the other player is doing the only stable outcomes are where on player withdraws and the other escalates.

These two outcomes, which are underlined in the matrix, are the only equilibria in this game. These equilibria are called the Nash equilibria, after the mathematician John Nash.

In an actual game of Chicken, if one player rips out his steering wheel in view of the other driver it visibly forecloses his choice to swerve. And hence the other player is forced to swerve. 

China is trying to convince India and the rest of the world that they do not have the option to swerve. Their position is that India has transgressed on their territorial sovereignty and they have no choice but to resort to an attack.

Many analysts, convinced by the Chinese rhetoric, assume that the Chinese will ultimately resort to an attack. Hence, India will be better off withdrawing immediately.

India, on the other hand has called for a mutual withdrawal. To achieve this, it needs to convince China that it has ripped out its steering wheel and will not swerve. 

When both players are convinced that the other will attack, the game will collapse to the (Stay, Stay) outcome. This means the conflict ends at the original Prisoner’s Dilemma game.

Then the two countries must cooperate to reach the mutually beneficial outcome of mutual withdrawal instead of needlessly expending resources in a futile standoff.

Also read: There is a simple solution to India-China Sikkim standoff, but it’s not easy

Writer

Vineet Gupta Vineet Gupta

The writer has a PhD in mathematics from Caltech, and has studied at Harvard, Stanford and Cambridge universities.

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