Learning Game theory the Sholay way…(are happy endings just Nash Equilibrium audience likes to see?)

What started as a series of Tweets has become a very interesting Mint column today.

Avinash Tripathi explains game theory basics using one of the iconic scenes of iconic movie Sholay:

This month Sholay will complete 42 years of its celluloid existence. As a Bollywood blockbuster and a cultural phenomenon, its enduring charisma needs explanation. Besides crisp dialogue, its deft portrayal of human emotions—from sublime loyalty to visceral revenge—has received critical acclaim. What has probably escaped the attention of critics is its portrayal of rational decision-making.

One sequence, in particular, stands out. The context of this scene is as follows: petty criminals Jai (Amitabh Bachchan) and Veeru (Dharmendra) are engaged by the former cop Thakur Baldev Singh (Sanjeev Kumar) to clean up the bandit-infested ravines of Ramgarh. In this ‘mission’, Veeru is infatuated by vivacious village belle Basanti (Hema Malini) who reciprocates his feelings.

The only problem in matchmaking is Basanti’s mausi (aunt), brilliantly played by veteran artist Leela Mishra, who dislikes Veeru due to his bad habits. Jai’s ill-fated and tragicomic interventions on behalf of his friend worsen the situation. To overcome mausi’s resistance, Veeru hatches a convoluted plan. He climbs atop a water tank and threatens to jump to his death unless his demand for Basanti’s hand in marriage is met. This peculiar setting reveals some subtle aspects of interpersonal decision making.

Avinash shows how Veeru’s threat was non-credible yet the irrational Mausi surrendered to the threat:

One way of making situations precise enough for exact analysis is to assign a numerical pay-off for each player for each possible outcome. While the exact numerical values are not particularly relevant for the discussion, two assumptions are critical.

First, Veeru prefers the outcome where he is married to Basanti and the aunt prefers the outcome where he is not. In a sense, this is the crux of the problem.

Second, both parties strongly dislike the outcome where Veeru actually jumps from atop the water tank, albeit due to different reasons. Veeru dislikes jumping because he may die or get injured (and Ramgarh probably doesn’t have decent medical facilities). The aunt dislikes this outcome due to empathetic distress, community pressure, and the possibility that she may have to grind mill in prison for abetting suicide (“chakki-pees-ing” as Veeru puts it).

In this game, the aunt, being the first mover, has two choices: acquiesce and agree to Basanti’s marriage with Veeru or hold her ground and play spoiler. Since Veeru is the follower, he can condition his action on his observations; his strategy must specify a plan of action for every possible contingency.

There are four logical possibilities: always jump; jump only if proposal is accepted; never jump; and jump only if the proposal is rejected. The last two strategies are particularly important.

A careful analysis of the outcome shown in the movie reveals something ingenious. Like mutually-supporting arches of a dome, choices made by the protagonists justify each other. Veeru’s threat justifies the aunt’s decision to accept the marriage proposal; otherwise she will have to bear the agonizing prospect of Veeru jumping from the tank. And the aunt’s decision justifies Veeru’s strategy. Since she agrees to the marriage proposal, he doesn’t have to carry out his costly threat and can walk away with his desired prize.

The configuration of choices that lead to this outcome is an example of what game theorists call Nash Equilibrium (NE), in honour of mathematician John F. Nash. Besides being a Nobel and Abel prize winner, Nash was also the subject of the Academy award winning movie A Beautiful Mind. Nash equilibria often have a yin-and-yang character that is portrayed beautifully in the film.

Superb. Nash Equilibrium was never this easier to figure.

If one put this in a 2 by 2 game it would be like this:

		Mausi
		Acquiesces	Rejects
Veeru	Jumps	Veeru would not do this unless he is so drunk that slips while coming down (which he almost does in the movie)	Veeru does not want to jump hoping Mausi takes his threat seriously
	Does not jump	This happens in the movie and is also the Nash Equilibrium	Mausi wants to reject but is unsure about Veeru’s threat

Though, there is not just one equilibrium.  What if Mausi is rational and rejects the threat:

The outcome depicted in Sholay is, however, not the only equilibrium. While the aunt’s choices, given her beliefs, are smart, her beliefs themselves are rather naive. Her decision to agree to the wedding critically depends on the expectation that if Veeru’s proposal is rejected, he will carry out his threat. But that is clearly irrational.

Veeru’s threat is a ‘non-credible threat’ that is not meant to be carried out. What if we require that not only her actions but also her beliefs (about the possibilities that may not materialize) are rational? We get another (more plausible) Nash equilibrium. In this case, Veeru never jumps from the tank, irrespective of the aunt’s decision. And the aunt rejects his marriage proposal. This is an example of what game theorists call ‘Subgame Perfect Nash Equilibrium’ (SPNE). One expects that most such dramatic events in real life end up with the outcome predicted by SPNE.

This would be something like this:

		Mausi
		Acquiesces	Rejects
Veeru	Jumps
	Does not jump		Sub Game Perfect Nash Equilibrium

Hmm..

What about linking games to movies?

Finally, one may wonder if applying an idea from game theory to a popular movie is a bit of a stretch. After all, it is unlikely that scriptwriters were aware of game theory. Even if they were, what is the logic behind choosing one abstruse solution concept over the other? The answer is that both scriptwriters and audience have an intuitive understanding of the relative likelihood of predicted outcomes.

An outcome that is not predicted by even a vanilla Nash equilibrium would have low chance of occurrence; as such, it will be seen as implausible deus ex machina. On the other hand, outcomes predicted by SPNE happen with high frequency in real life and seem routine and boring.

NE predicts precisely the outcome that is both plausible and interesting. Subconsciously it generates an ‘ah ha!’ moment for the audience. Speculatively, this cinematic logic must have been decisive.

“Is story mein emotion hai, drama hai, tragedy hai” said the protagonist with typical rhetorical flourish. He could have well added that besides emotion, drama and tragedy, the story also contains strategic manipulation, game theory, and plenty of fun.

Hmmm.. Sholay indeed had everything in it.

It seems happy endings depicted in so many of our movies are quite like the Nash Equilibrium. Despite many sequences and changes, eventually most show happy endings. We don’t see the heroine walking off with the villain. Very few show that hero & heroine both go in different ways.

Taking Sholay’s case, say Mausi rejects and Veeru jumps. Would people have accepted this shock?

For instance, one’s mind goes to Baazigar where Shahrukh the Hero throws Shilpa Shetty from the terrace of a building. This building housed the Registered Marriage office and just a scene ago both were supposed to get married but the office closed due to lunch. And just a scene later the Hero does the opposite. This is expected from villains but not heroes. The scene shocked audience but they accepted it and the risk SRK took worked brilliantly.

Likewise, there are some movies which do the unusual and wait for audience reaction. If it is a aha the movie is a hit otherwise it just flops.

It will be really interesting to think through many of these games via more such movies/well-known instances. After all game theory becomes more interesting this way…