The number of correct guesses with partial feedback (original) (raw)

No-feedback Card Guessing Game: Moments and distributions under the optimal strategy

2022

Relying on the optimal guessing strategy recently found for a no-feedback card guessing game with k-time riffle shuffles, we derive an exact, closed-form formula for the expected number of correct guesses and higher moments for a 1-time shuffle case. Our approach makes use of the fast generating function based on a recurrence relation, the method of overlapping stages, and interpolation. As for k > 1-time shuffles, we establish the expected number of correct guesses through a self-contained combinatorial proof. The proof turns out to be the answer to an open problem listed in Krityakierne and Thanatipanonda (2022), asking for a combinatorial interpretation of a generating function object introduced therein.

The card guessing game: A generating function approach

Journal of Symbolic Computation

Consider a card guessing game with complete feedback in which a deck of n cards ordered 1,. .. , n is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at a time under the optimal strategy until no cards remain. We provide an expression for the expected number of correct guesses with arbitrary number of terms, an accuracy improvement over the results of Liu (2021). In addition, using generating functions, we give a unified framework for systematically calculating higher-order moments. Although the extension of the framework to k ≥ 2 shuffles is not immediately straightforward, we are able to settle a long-standing conjectured optimal strategy of Bayer and Diaconis (1992) by showing that the optimal guessing strategy for k = 1 riffle shuffle does not necessarily apply to k ≥ 2 shuffles. Substituting this ansatz in the recurrence relation, we end up with system of linear equations for the coefficients a 1 , a 2 ,. .. , which can subsequently be used to solve for as many coefficients as we wish.

A variant of the guessing secrets game

2005

In [2], Chung et al. introduced the Guessing Secrets game. In [4] the authors introduce a variant of this game in which for each question the reference secret is chosen at random (with uniform distribution) by Responder. In this paper we investigate another variant in which Responder is required to answer truthfully to questions of the form How many secrets are there in X?, where X is a subset of the search space chosen by Questioner. We investigate the cost (number of questions needed) in the worst case, in the best case and in the average case.

No Feedback? No Worries! The art of guessing the right card

2022

In 1998, Ciucu published "No-feedback card guessing for dovetail shuffles", an article which gives the optimal guessing strategy for n cards (n even) after k riffle shuffles whenever k > 2 log 2 (n). We discuss in this article the optimal guessing strategy and the asymptotic (in n) expected number of correct guesses for any fixed k ≥ 1. This complements the work achieved two decades ago by Ciucu.

A New Variation of Hat Guessing Games

Lecture Notes in Computer Science, 2011

Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of n players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least k correct guesses and no wrong guess for the players to win the game, but they can choose to "pass". A strategy is called perfect if it can achieve the simple upper bound n n+k of the winning probability. We present sufficient and necessary condition on the parameters n and k for the existence of perfect strategy in the hat guessing games. In fact for any fixed parameter k, the existence of perfect strategy can be determined for every sufficiently large n. In our construction we introduce a new notion: (d1, d2)-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the kdominating set of the hypercube. It also might be interesting in coding theory. The existence of (d1, d2)-regular partition is explored in the paper and the existence of perfect k-dominating set follows as a corollary.

Guessing Games on Undirected Graphs

2015

Guessing games for directed graphs were introduced by Riis for studying multiple unicast network coding problems. In a guessing game, the players toss generalised die and can see some of the other outcomes depending on the structure of an underlying digraph. They later simultaneously guess the outcome of their own die. Their objective is to find a strategy that maximises the probability that they all guess correctly. The performance of the optimal strategy for a digraph is measured by the guessing number. In general, the existence of an algorithm for computing guessing numbers of a graph is unknown. In the case of undirected graphs, Christofides and Markström defined a strategy that they conjectured to be optimal. One of the main results of this thesis is a disproof of this conjecture. In particular, we illustrate an undirected graph on 10 vertices having guessing number which is strictly larger than the lowerbound provided by Christofides and Markström’s method. Moreover, even in c...

Games with Imperfect Information

1993

An information set in a game tree is a set of nodes from which the rules of the game require that the same alternative (i.e., move) be selected. Thus the nodes an information set are indistinguishable to the player moving from that set, thereby reflecting imperfect in- formation, that is, information hidden from that player. Information sets arise naturally in (for example) card gaines like poker and bridge. IIere we focus not on the solution concept for im- perfect information games (which has been studied at length), but rather on the computational aspects of such games: how hard is it to compute solutions? We present two fundainental results for imperfect informa- tion games. The first result shows that even if there is only a single player, we must seek special cases or heuristics. The second result complements the first, providing an efficient algorithm for just such a special case. Additionally, we show how our special case algo- rithm can be used as a heuristic in the general...

Guessing games with homogeneous and heterogeneous players: An experimental reconsideration

2004

Abstract We replicate an experiment previously reported in this journal (Güth, Kocher and Sutter 2002). Our results are at variance with their results, but confirm their key hypothesis that heterogeneous players guess closer to the equilibrium than homogeneous players. Keywords: Guessing Game, experiment, levels of reasoning Classification code: C72, C91, C92

On the optimal strategy in a random game

Electronic Communications in Probability, 2004

Consider a two-person zero-sum game played on a random n × n-matrix where the entries are iid normal random variables. Let Z be the number of rows in the support of the optimal strategy for player I given the realization of the matrix. (The optimal strategy is a.s. unique and Z a.s. coincides with the number of columns of the support of the optimal strategy for player II.) Faris an Maier [4] make simulations that suggest that as n gets large Z has a distribution close to binomial with parameters n and 1/2 and prove that P (Z = n) ≤ 2 −(k−1). In this paper a few more theoretically rigorous steps are taken towards the limiting distribution of Z: It is shown that there exists a < 1/2 (indeed a < 0.4) such that P (1 2 − a)n < Z < (1 2 + a)n → 1 as n → ∞. It is also shown that EZ = (1 2 + o(1))n. We also prove that the value of the game with probability 1 − o(1) is at most Cn −1/2 for some C < ∞ independent of n. The proof suggests that an upper bound is in fact given by f (n)n −1 , where f (n) is any sequence such that f (n) → ∞, and it is pointed out that if this is true, then the variance of Z is o(n 2) so that any a > 0 will do in the bound on Z above.

C O ] 9 O ct 2 01 4 Guessing Games on Triangle-free Graphs

2014

The guessing game introduced by Riis[15] is a variant of the “guessing your own hats” game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markström [6] introduced a method to bound the value of the guessing number from below using the fractional clique number κf (G). In particular they showed gn(G) ≥ |V (G)| − κf (G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman–Sims graph has guessing number at least 7...