Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω(k2), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O(k.log k/ϵ2) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O(k.log k/ϵ4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O(k.log k/ϵ4) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria

Approximate Well-supported Nash Equilibria below Two-thirds

In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing his behaviour. Recent work has addressed the question of how best to compute epsilon-Nash equilibria, and for what values of epsilon a polynomial-time algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most epsilon less than the best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new technique that leads to an improvement to the worst-case approximation guarantee

What do voters want from an online voting experience?

The UK does not offer remote online voting now. But, as digital services and digital citizen-government interactions continue to grow, voting could be offered remotely online (i-voting) in future alongside postal and in-person ballots. So what might i-voting look like, and how might voters respond? What makes an i-voting experience positive or negative? We designed a prototype voting app that mirrors the traditional ballot paper. We asked potential voters to use it, on 3 different digital devices, and compared it with the in-person experience. Our study involved a diverse sample of 32 people from the Brunel community who kindly gave their time to test the alternative voting mode and share their feedback and opinions with us. We are grateful for their participation. Our key findings are: • Gaining first-hand experience of the app was associated with either maintaining or improving willingness to vote online – the ‘maintainers’ generally had a high initial willingness to vote online, while the ‘improvers’ were initially less keen. • If i-voting were to become an option in future, our respondents want to see stronger security and authentication features on an i-voting app, education to inform voters about how i-voting works, and transparency about data risks, actors involved, and the security measures in place to prevent fraud and malpractice. Further testing, on a larger scale, could usefully explore which voter groups might benefit most from the option of i-voting, and how design features and voter engagement could address security and data protection concerns.Brunel University London