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1. Inverse Matrix Games With Unique Quantal Response Equilibrium

   https://doi.org/10.1109/LCSYS.2022.3214857  

   Yu, Yue ; Salfity, Jonathan ; Fridovich-Keil, David ; Topcu, Ufuk ( January 2023 , IEEE Control Systems Letters)
2. On non-unique solutions in mean field games

   https://doi.org/10.1109/CDC40024.2019.9029906  

   Hajek, Bruce ; Livesay, Michael ( December 2019 , 2019 58th Conference on Decision and Control)

   The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $\epsilon$-Nash Markov equilibria for $N$ players with $\epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping. 

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3. High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

   https://doi.org/10.1137/1.9781611977073.47  

   Bafna, Mitali ; Hopkins, Max ; Kaufman, Tali ; Lovett, Shachar ( January 2022 , Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight ℓ2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof for the former ℓ2-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, where in many cases we improve the state of the art (Barak, Raghavendra, and Steurer FOCS 2011, and Arora, Barak, and Steurer JACM 2015) from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the q-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an ℓ∞-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture (Khot, Minzer, and Safra FOCS 2018). We give a reduction from a related ℓ∞-variant to our ℓ2-characterization, but it loses factors in the regime of interest for hardness where the gap between ℓ2 and ℓ∞ structure is large. Nevertheless, our results open the door for further work on the use of HDX in hardness of approximation and their general relation to unique games. 

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4. Approximating Cumulative Pebbling Cost Is Unique Games Hard

   https://doi.org/10.4230/LIPIcs.ITCS.2020.13  

   Blocki, J ; Lee, S ; Zhou, S. ( January 2020 , Leibniz international proceedings in informatics)

   null (Ed.)

   The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = min_P ∑_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of G and |P_i| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) f_{G,H} [Joël Alwen and Vladimir Serbinenko, 2015] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(⋅). A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find x such that f_{G,H}(x) = h. Thus, to analyze the (in)security of a candidate iMHF f_{G,H}, it is crucial to estimate the value cc(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is NP-Hard to compute cc(G), but their techniques do not even rule out an efficient (1+ε)-approximation algorithm for any constant ε>0. We show that for any constant c > 0, it is Unique Games hard to approximate cc(G) to within a factor of c. Along the way, we show the hardness of approximation of the DAG Vertex Deletion problem on DAGs of constant indegree. Namely, we show that for any k,ε >0 and given a DAG G with N nodes and constant indegree, it is Unique Games hard to distinguish between the case that G is (e_1, d_1)-reducible with e_1=N^{1/(1+2 ε)}/k and d_1=k N^{2 ε/(1+2 ε)}, and the case that G is (e_2, d_2)-depth-robust with e_2 = (1-ε)k e_1 and d_2= 0.9 N^{(1+ε)/(1+2 ε)}, which may be of independent interest. Our result generalizes a result of Svensson who proved an analogous result for DAGs with indegree 𝒪(N). 

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5. A periodic isoperimetric problem related to the unique games conjecture

   https://doi.org/10.1002/rsa.20877  

   Heilman, Steven ( July 2019 , Random Structures & Algorithms)

   We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Letn ≥ 2. Suppose a subset Ω ofn‐dimensional Euclidean spacesatisfies −Ω = Ω^cand Ω + v = Ω^c(up to measure zero sets) for every standard basis vector. For anyand for anyq ≥ 1, letand let. For anyx ∈ ∂Ω, letN(x) denote the exterior normal vector atxsuch that ‖N(x)‖₂ = 1. Let. Our main result shows thatBhas the smallest Gaussian surface area among all such subsets Ω, less a small error:In particular,Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10⁻⁹would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.

    

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