Monday, 9 January 2017

The game theory of withholding supply

Back in October, Ford stopped producing the Falcon XR8 Sprint. What happened? According to the Daily Telegraph:
FORD dealers are charging a staggering $30,000 more than the recommended retail price — up from $60,000 to $90,000 — for the final Falcon V8 sedans as buyers try to secure a future classic.
The last batch of Falcon V8s was thought to have sold out, but some dealers held a secret stash to release them onto the market in the final days of production so they could jack up the price.
Why did the price rise? Ford Australia boss Graeme Whickman explains:
“It’s supply and demand,” said Mr Whickman. “We set a wholesale price and recommended retail price … but at the end of the day the dealer and the customer decide what the vehicle is going to be sold and bought for,” he said, referring to high prices being charged for the initial shipment of Mustangs last year.
Demand for the Falcon XR8 was high, and the sellers withheld some supply - high demand and lower supply ensures that prices will increase. in this case by up to 50%.

So, why don't firms do this all the time? Why not withhold supply all the time? A bit of game theory can help, as laid out in the payoff table below [*]. The seller can choose to withhold stock, or not. The buyer can choose to buy now, or wait and buy later.


Where is the Nash equilibrium in this game? Consider the seller's choice first. If the buyers choose to buy now, the seller is better off choosing to withhold some stock, because profits will be higher. If the buyers choose to wait and buy later, the seller is better off choosing to not withhold stock, because profits will be higher. So, the seller doesn't have a dominant strategy (a strategy that is always better for them, no matter what the buyers choose to do).

Now consider the buyers' choice. If the seller chooses to withhold some stock, the buyers are better off choosing to wait and buy later, since they will force the price to fall when the withheld stock is released.  If the seller chooses not to withhold stock, the buyers are better off to buy now, or they will miss out on a car. So, the buyers don't have a dominant strategy either.

Neither player has a dominant strategy, so what is the solution to this game? We can find it using the best response method (which we've already described in the last two paragraphs). Any combination of strategies where both players are choosing their best response to the other player's strategy is a Nash equilibrium. However, in this case there is no Nash equilibrium (at least, no equilibrium in pure strategy). Instead, there will be a mixed strategy equilibrium where both the seller and the buyers should randomise their actions. The seller should sometimes withhold stock, and other times not.

Of course, that analysis assumes the game is not repeated. Car sellers sell new models each year, so really this is a repeated game. How does this change the game? If a game is repeated, then players can develop reputations. So, this actually reinforces the importance of the seller not constantly withholding stock. If they develop a reputation for withholding stock to sell later, then buyers will recognise that the seller's prices are artificially high and will wait until the seller releases the withheld stock, lowering the price.

So, provided buyers (as a group) learn from this experience, then there is little to fear from Ford dealers repeating the exercise every time a popular car is withdrawn from production. The problem is, of course, that it is unlikely to be the same group of buyers next time around.

*****

[*] This game is presented as a simultaneous game, even though the seller clearly chooses their strategy before the buyer. This is because when the buyer chooses to buy now or wait, they don't know whether the seller has withheld stock or not. For simplicity, we're also assuming that all buyers act the same way, when of course they won't. However, the overall point still stands even if we have multiple buyers in the game.

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