The complexity of approximating a trembling hand perfect equilibrium of a multi-player game in strategic form

The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game of perfect recall

“…(We will later formally define FIXP a , as well as its real-valued progenitor FIXP, and the piecewise-linear fragment linear-FIXP (= PPAD).) Very recently, in a paper with Hansen, Miltersen, and Sørensen [11], building on [12], we have shown that for NFGs with n ≥ 3 players, approximating a "trembling-hand perfect equilibrium" (PE) within desired precision is also FIXP a -complete. Since PEs refine NEs, we only had to show containment in FIXP a .…”

“…Also, by contrast to [9] we make no use of reductions to graphical games. Instead, we combine older insights, including Kuhn and Selten's original agent normal form for EFGPRs, and Myerson's alternative definition of PE using ǫ-PEs (both for normal and extensive form), with recently developed fixed point functions for equilibria of n-player normal form games, n ≥ 3, developed in [12] and [11].…”

“…More specifically, a key to our results is this: in Section 3, we adapt a construction in [11] of a fixed point function for "ǫ-PEs" of a given NFG (which itself is an adaptation of a fixed point function for NEs of NFGs given in [12]) to show that to any n-player EFGPR, G, we can associate a continuous function F ǫ G (x), defined by a "small" algebraic circuit over {+, * , max} (whose encoding size is polynomial in that of G), where ǫ in an input parameter to the circuit, and such that, for any fixed ǫ > 0, the function F ǫ G (x) maps the space of behavior strategy profiles of G to itself, such that the Brouwer fixed points of F ǫ G (x) constitute ǫ-PEs of G. This proves that computing an ǫ-PE, given G, ǫ , is in FIXP, even when ǫ > 0 is given succinctly by an algebraic circuit.…”

“…Even though we can not construct the agent normal form explicitly (because it is exponentially large), it turns out that we do not need to: by combining these various facts, we can nevertheless construct a "small" algebraic circuit for F ǫ G (x), by adapting the analogous construction from [11]. With the functions F ǫ G (x) (and H ǫ G (x)) in hand, in Section 4 we then use (similar to [11]) algebraic circuits to construct a "very very small" ǫ * > 0 for which we can prove, using results from real algebraic geometry ( [33,2]), that every fixed point of F ǫ * G (x) is δ-close (in ℓ ∞ ) to an actual PE. Likewise, we show that every fixed point of H ǫ * G (x) is δ-close to a QPE.…”

“…Also, our proof of FIXP a -completeness for computing a QPE involves a novel fixed point characterization. By comparison to these, our proof of FIXP acompleteness for PE is technically easier, given the prior results in [11,12], and given long existing results in the literature on EFGPRs which we exploit. Potential computational applications.…”