The Dishoom Dice

This story is about an Indian restaurant in London called ‘Dishoom’, which has ingeniously applied probability theory and economics to its business model.  

Dishoom found that while they are generally booked out, they tend to have vacant tables before 6pm, Monday to Thursday. So, they came up with a unique loyalty program in which members who pay their bill before 6pm, Monday to Thursday, are given a die to roll at the end of their meal; if a 6 is rolled, the meal is free. Apart from the maths (discussed below), the idea is clever because of its simplicity and the buzz that it has been able to create.

 

Here's the maths.  

 

For the customer, there’s the excitement of the possible free meal and the luck of the roll – a 1 out of 6 chance that the meal will be free. However, the mathematical analysis is as follows. Assuming a meal cost of £60, the customer’s expected saving is (1/6 × 60) + (5/6 x 0) = £10 per the expected value formula (mean of the probability distribution), which is μ = ΣpX. Mathematically, this is just a 16.7% discount, and if Dishoom simply said that members would get a 16.7% off their meal, they’d probably just shrug, but telling them, “Roll a 6 and your meal is free,” has got everyone excited. Customers are subconsciously playing this mini lottery, attracted by even the low 1/6^th chance of getting a free meal-  I learnt that economists call this the ‘prospect bias’, where people overvalue the small probability given the big reward. 

 

For Dishoom on the other hand, it is a win-win situation. Firstly, it has got publicity from its idea - winners post about their free meals and even losers talk about how close they were. The restaurant is now fuller during its quieter times.

 

For every additional table reservation under this offer the expected revenue for Dishoom is £60-£10 = £50. This is the converse of the £10 expected value calculation for the customer.  If Dishoom now fills 10 tables that were going vacant before, it earns an additional £500 per day on average, which works out to £2000 pounds a week (Monday to Thursday). You can also get this result by using the formula P(X) = Σ(nCr)pX, where nCr represents the number of combinations in which a specific number of people (r) can get the discount, p represents the probability of that number of people getting the discount and X represents the earning for Dishoom if r people get the discount [see detailed working in table below].

 

Most interestingly, Dishoom ends up billing more per table because people are spending more than they otherwise would – thinking that since the meal might be free, they might as well order one more dessert!

 

Dishoom’s additional earnings due to customers ordering more is 5/6 x a, where a represents the additional amount customers spend on ordering more food in the hope of getting their meal free and making the most out of the deal. Ironically, while the  consumers feel luckier, it is really Dishoom that stands to earn an additional £5 per bill (5/6 x 10% x 60), assuming customers spend 10% more, and continuing with the earlier assumption that the average bill is £60.

 

The genius of this idea is that no customer looks at this offer and thinks of it as a 16.7% discount. A professional poker player or a mathematician might, but most people fall for the prospect bias. As economists explain, people think in moments, not rationally in expected values. Rolling a six feels like defying probability, and Dishoom has turned that feeling into a small, beautiful trap: the more you try to maximise your win, the more they actually win.

Working

r	p	nCr	(nCr)p	X	(nCr)pX
0	0.334898	1	0.334898	360	120.56
1	0.066980	6	0.401878	300	120.56
2	0.013396	15	0.200939	240	48.23
3	0.002679	20	0.053584	180	9.65
4	0.000536	15	0.008038	120	0.96
5	0.000107	6	0.000643	60	0.04
6	0.000021	1	0.000021	0	-
			1.000000		300.00