Opponent Wins Research Articles

Classroom Experiments of Auctions

A few non-computerized classroom auctions of an indivisible object were presented and analyzed. Two of the auctions were played in a classroom of around thirty students. Both auctions have big possibilities of various bids being equal, the effect of which is that, the played Vickery auction is revenue-equivalent to a quite uncommon variation of the ascending-bid auction where defaulting and the auctioneer's calling strategy play vital roles. I examine their revenue-equivalences based on two slightly modified intuitive assumptions of Haile and Tamer: bidders neither risk to pay more than their valuations nor risk to let an opponent win at a price less than or equal to their valuations. The results of the experiments demonstrate the inconsistency of the students' goals.

Inference with an Incomplete Model of English Auctions

While English auctions are the most common in practice, their rules typically lack sufficient structure to yield a tractable theoretical model without significant abstractions. Rather than relying on one stylized model to provide an exact interpretation of the data, we explore an incomplete model based on two simple assumptions: bidders neither bid more than their valuations nor let an opponent win at a price they would be willing to beat. Focusing on the symmetric independent private values paradigm, we show that this limited structure enables construction of informative bounds on the distribution function characterizing bidder demand, on the optimal reserve price, and on the effects of observable covariates on bidder valuations. If the standard theoretical model happens to be the true model, our bounds collapse to the true features of interest. In contrast, when the true data‐generating process deviates in seemingly small ways from that implied by equilibrium in the standard theoretical model, existing methods can yield misleading results that need not even lie within our bounds. We report results from Monte Carlo experiments illustrating the performance of our approach and comparing it to others. We apply our approach to U.S. Forest Service timber auctions to evaluate reserve price policy.

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f 1, f 2,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f i . It is proved that there exist constants c= c( n) and k 0= k 0( n) such that f k+1 = f k + f k− c for all k> k 0, and lim n→∞ c(n)/(n logn)=1 .

Geography

Generalized Geography is an impartial two-person game played on a digraph G=( V, A). In impartial Arc (Vertex) Geography, a token is initially placed on a special start vertex, and the players alternately move the token along unused arcs (vertices) of G. The player first unable to move loses and his opponent wins. The question of who wins these games IAG and IVG is known to be PSPACE-complete. Both impartial versions with two tokens on special start vertices are proved PSPACE-complete even for DAGs but polynomial for directed trees. The partizan variations, PAG and PVG, with one token per player are PSPACE-complete even for bipartite degree-restricted digraphs. They are NP-hard for DAGs, but polynomial for directed trees.

Epidemiography with various growth functions

Epidemiography designates a class of combinatorial games played on directed graphs. At step k of the game, the move made on a digraph G is replicated onto f(k) isomorphic copies of G. The player first unable to move loses and his opponent wins. Let G be any finite acyclic digraph. We determine the winner for several classes of growth functions f and show that in each case the winner has a semi-robust strategy: He can play almost randomly during the play except at the end. We also determine the strategy for shortest and longest play.

Geodetic contraction games on graphs

Let G = (V, E) be a finite graph. The set L of labeled vertices is initially empty. Two play- ers A and B move alternately (A first), by choosing an unlabled vertex u C V L; then u itself and all vertices on shortest paths between u and any vertex of L are adjoined to L. When L = V, the game is over. In normalplay, the first player unable to move loses and his opponent wins. The out- come is reversed for misbreplay. We resolve the game by determining its winning strategies for the following cases: trees in normal play; cycles in normal and misbre play; and complete graphs K m with rays all of length n, in normal play.

Properties of K-Welter's game

Place n coins on distinct squares of a semi-infinite strip. The squares are numbered with the nonnegative integers 0,1,2,3,… from the left end of the strip. Each of the two players alternately moves a coin to a lower unoccupied square, at most k squares from its present position, where k is a fixed positive integer. The player first unable to move (because the coins are jammed in the lowest possible numbered squares 0,1,2,..., n−1) loses and his opponent wins. The main interest of this paper is in investigating interesting properties of the game, presenting a solution for the case of three coins(> n=3) for any k (by giving a strategy with a polynomial algorithm) and investigating properties of the Sprague-Grundy function for this case, including its polynomial computation for the subcase k=2 m −2.

Applications: Digraphs, Matrices, and Football-Team Ratings

In Ohio a statewide rating system known as the Harbin Football-Team Rating System determines the high school football teams that are eligible to compete in postseason playoffs. Each team earns points for games it wins and for games that a defeated opponent wins. The teams with the most points “go to state.”

K-welter—a generalization of Welter's game

Finitely many coins are placed on distinct squares of a semi-infinite linear board ruled into squares numbered 0, 1, 2,… Each of the two players alternately moves a coin to a lower unoccupied square, at most k squares from its present position, where k is an arbitrary fixed positive integer. The player first unable to move loses and his opponent wins. We give complete solutions for the case of two coins for any k; and for any number of coins when k = 2 m − 1 ( m ⩾ 1). We also prove that, for any k, the Sprague-Grundy function of the game is invariant under translations of coins by k + 1.

Partizan octal games: Partizan subtraction games

A subtraction gameS=(s1, ...,sk)is a two-player game played with a pile of tokens where each player at his turn removes a number ofm of tokens providedmeS. The player first unable to move loses, his opponent wins. This impartial game becomes partizan if, instead of one setS, two finite setsSL andSR are given: Left removes tokens as specified bySL, right according toSR. We say thatSL dominatesSR if for all sufficiently large piles Left wins both as first and as second player. We exhibit a curious property of dominance and provide two subclasses of games in which a dominance relation prevails. We further prove that all partizan subtraction games areperiodic, and investigatepure periodicity.

On the asymptotic normality of the number of replications of a paired comparison

Consider the number Xm of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.

On the asymptotic normality of the number of replications of a paired comparison

Consider the number Xm of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.

Asymptotic properties of the number of replications of a paired comparison

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

Asymptotic properties of the number of replications of a paired comparison

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

Learning Games through Pattern Recognition

The objective of this research was to investigate a technique for machine learning that would be useful in solving problems involving forcing states. In games or control problems a forcing state is one from which the final goal can always be reached, regardless of what disturbances may arise. A program that learns forcing states in a class of games (in a game-independent format) by working backwards from a previous loss has been written. The class of positions that ultimately results in the opponent's win is learned by the program (using a specially designed description language) and stored in its memory together with the correct move to be made when this pattern reoccurs. These patterns are searched for during future plays of the game. If they are formed by the opponent, the learning program blocks them before the opponent's win sequence can begin. If it forms the patterns first, the learning program initiates the win sequence. The class of games for which the program is effective includes Qubic, Go-Moku, Hex, and the Shannon network games, including Bridge-it. The description language enables the learning program to generalize from one example of a forcing state to all other configurations that are strategically equivalent.