Game Theory & Prisoner’s Dilemma

Prisoner’s Dilemma

The best and most famous introduction to Game Theory involves the Prisoner’s Dilemma problem.

The game goes like this:

  • Two criminals have been caught, and are being kept in separate interrogation rooms.
  • The police are asking them each to confess.
  • If neither of them confess, the case against them will be weak, and they will each get 2 years jail.
  • If one of them confesses and the other doesn’t, the confessor will get 1 year, whilst the other will get 10 years jail.
  • If they both confess, they will get 5 years each, because after all – they were guilty!

What would you do, in their situation? We can put this in a matrix to better illustrate the moves and payoffs.

By reading ‘down the column’ of Prisoner B’s move, we can find the best course of action for Prisoner A.

  • If Prisoner B confesses, then Prisoner A is better off confessing (5 years vs 10 years)
  • If Prisoner B doesn’t confess, then Prisoner A is better off confessing (1 year vs 2 years)

Thus, regardless of what the other party does, Prisoner A gets the best outcome by confessing. This is known as a dominant strategy.

Let us now pretend we have a great relationship with Prisoner B, and we really would like to consider ‘not confessing’. If both parties don’t confess, we can get away with 2 years each. However, can we trust Prisoner B?

Whilst Prisoner A might be considering not confessing, what does this look like from Prisoner B’s perspective?

If they risk ‘not confessing’, whilst Prisoner A confesses, they’ll be doing 10 years hard time!

The same dominant strategy exists for Prisoner B also; they get a better deal no matter what Prisoner A does by confessing.

This is the heart of the Prisoner’s Dilemma. That the two parties could achieve an optimal solution for both (4 years total time), if only they could be trusted. Game Theory says that hoping your counterpart will act in good faith is not a strategy. Parties can only be trusted to act in their own self-interest.

The Product Manager’s Dilemma

How then does this scenario play out in a software context?

Let us say that instead of Prisoners, we have two Software Companies: A & B. They each offer similar products and together form a duopoly.

The products in this case are quite similar, and thus price is a determining factor in buyer’s choice. For simplicity, we will assume that all new customers this year will select *exclusively* on price.

It is the start of a new financial year, and the Companies need to reveal their new pricing. The size of this market is 1,000 users.

  • If Company A & B both set their licenses to $200 per user per month, they will each capture 50% of the market.
    • Payoff for Company A: 500 * $200 = $100,000 MRR
    • Payoff for Company B: 500 * $200 = $100,000 MRR
  • If Company A sets their price at $100, and Company B sets their price at $200, Company A will capture all of the new Users.
    • Payoff for Company A: 1,000 * $100  = $100,000
    • Payoff for Company B: 0 * $200 = -$0.00 $0
  • If Company A sets their price at $200, and Company B sets their price at $100, Company B will capture all of the new Users.
    • Payoff for Company A: 0 * $200 = $0
    • Payoff for Company B: 1,000 * $100  = $100,000
  • If Company A & Company B set their licenses to $100 per user per month, they will each capture 50% of the market:
    • Payoff for Company A: 500 * $100 = $50,000 MRR
    • Payoff for Company B: 500 * $100 = $50,000 MRR

How would you price in this case?

You can see again we have a Prisoner’s Dilemma. Both companies can make double the revenue by keeping their prices high. However unlike the Prisoner’s – which may have had a good relationship – the competing Companies certainly want to maximise their own profits. By cutting price, they can capture the lion’s share of the market. It is simply too rewarding to ‘cheat’ on the ‘collusion’ price (putting aside the fact it is illegal).

In fact, price fixing via cartels is often fundamentally fragile for this very reason (Pindyck and Rubinfeld, 2012). The first turncoat will inevitably profit in a large way.

So let us say that both parties price at $100 this year. How would you price next year?

If a Company were to price at $99, they would again capture the entire market. Despite the lower cost, this would be better than sharing the volume 50/50.

If it makes sense for Company A to price at $99, then it makes sense for Company B to price at $98.

And so on.

This is how price wars start. With the Game Theory lens, you can see the logic behind them. The question is, when will it end?

According to economic theory, prices can go down to their ‘marginal cost’. Marginal cost is effectively the cost to product an additional unit. At this price you gain no surplus, nor do you make any profit. However given you’d be losing money if you didn’t go down that low, you have little choice. You stop at that point, because you’d lose money on each sale if you sold below marginal cost.

For this game, we’ll assume both companies had a margin cost of $30 per user per seat, to cover maintenance and hosting. Each Company priced as such, gained 50% market share, and made 500 * $30 = $15,000 MRR. This is a far cry from the $100K ‘price collusion’ number!

Now let us setup a second software game.

In this game, two new Product Managers have been hired at each company. They are given the historical pricing information as follows:

  • Company A’s price was $80 per user per month
  • Company B’s price was $80 per user per month

At 50/50 market share, each company is making 500 * $80 = $40,000 MRR.

In this case, their products are not identical.

In fact, 50% of the market would pay $40 more for Company A’s product.

And 50% of the market would pay $40 more for Company B’s product.

Given this, how would you price your product?

Let’s evaluate Company A’s options, based on Company B’s possible moves: $80, $40, and below $40.

  • At $80, Company would be best to stay at $80 too, each sharing half the market.
  • At $40, Company A would be best to stay at $80, each sharing half the market.
  • At <$39, it becomes more complex.
    • To stay at $80 would be foolish; they would lose half the market.
    • Moving down to $79 would make sense; they could keep half the market, at the highest price possible.

Given that Company A could keep moving down, staying at $40 above Company B, does it make sense that Company B move from $80?

Company B suddenly has to price cut a lot simply to win the market at all. Even if they price cut to $39 and won the whole market, 1,000 * $39 = $39,000. This is less then they were getting at 500 * $80 = $40,000.00.

We have hit a point where it doesn’t make sense for either player to move for this initial position. This is known as the “Nash Equilibrium”.

We can now trust that neither player will move on price, because it’s not in their interest to do so.

They are also locked into a higher price at $80, than they were in the perfectively competitive Game 1, where a price war lowered it to $30.

What then prevented this price war and held prices high?

Willingness to pay, through differentiation.

We see many examples of this in the marketplace. In the airline market, willingness to pay (and therefore prices) are held high due to frequent flyer points and loyalty programs. For any given airline to steal market share from the other airlines, they need to drastically cut prices to compensate for the lack of transferrable points. However that cost-cutting exercise means they get less income from their original customers too!

It also explains why Apple and Samsung have been able to keep prices on their phones high. Through ecosystems including apps & media, consumers have an in-built switching cost. Some Apple customers might pay $100, $200 or even $300 more to ‘stay Apple’, rather than losing access to their apps, media & photos. The same can be said for Android customers, and thus neither company has great incentive to cut prices. They are in a Nash Equilibrium.

Price War vs Features War

Pricing aside, features can affect purchasing intent. You can replace the above pricing decisions with one about building a given feature; let’s say ‘PDF Export’ for a CRM program.

If none of the major products built a ‘PDF Export’ feature, they would save resources, and the customers would still buy their software. However if just one of the major players does in fact build this feature, they may take the lion’s share of the market. Thus, companies may feel compelled into a ‘feature arms race’; a bizarro-world Product version of a price war, bloating their products with

This presents something of a paradox. To prevent price wars, you need to build differentiation. However you may lose out to richer feature sets, if you don’t engage in the feature war that your competitors do. How then to proceed?

The keys are to both know where you suck and where it matters, and how to build non-functional differentiation.

These are topics covered in the following chapters.

References

Pindyck, R. and Rubinfeld, D. (2012). Microeconomics. 8th ed. Boston: Pearson.