Lecture 7 -- Privacy Preserving Public Information for Sequential Games

Our seventh lecture was given by Jamie Morgenstern, about her very interesting paper joint with Avrim Blum, Ankit Sharma, and Adam Smith.

The birds-eye view is the following:

Suppose players (think financial institutions) take turns sequentially deciding which investments to make, from amongst a feasible set, which can be different for each player. In general, the profit that a player gets from an investment is a decreasing function of how many players previously have made the same investment. (Perhaps these investments are structured like pyramid schemes, where there is a big advantage in getting in early, or perhaps the market is somehow destabilized if there is too much capital invested in it).

We can study this interaction in various information models. In the complete information setting, each player sees exactly how much has been invested in each resource at the time that she arrives, and the unique dominant strategy solution of the game is for each player to invest greedily. They show that this solution achieves a good constant factor approximation to the optimal welfare.

But if the players are financial institutions, then their investments might represent sensitive trade secrets, and they may not want to share this information with others -- in which case the complete information setting seems unrealistic. This could be very bad news however -- if players have no information at all about what has gone on before their arrival, its not hard to cook up plausible sounding behaviors for them which result in disastrous welfare outcomes.

So the paper asks: can we introduce a differentially private signal (so one that necessarily reveals little actionable information about each agent, and therefore one whose introduction the agents have little reason to object to) that nevertheless allows the market to achieve social welfare that approximates OPT.

Skipping over some details, this paper shows that the answer is yes. Making public a differentially private count of how much has been invested in each resource as the game plays out is enough to guarantee that sequential play (studied either as the simple behavioral strategy in which players imagine that the noisy counts are exactly correct, or any solution in which players play only undominated strategies) results in an outcome that has a bounded competitive ratio with OPT.

This paper also contains an interesting technique that will probably be useful in other contexts: they develop a new method of privately maintaining the count of a set of numbers that achieves better additive error as compared to previous work, at the cost of introducing some small multiplicative error. In the application they need counters for in this paper, this modification gives improved overall bounds.