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A Study of Quantum Strategies for Newcomb's Paradox

Takashi Mihara

Published Jan 1, 2010• 0 citations
Publisher

Abstract

Newcomb’s problem is a game between two players, one of who has an ability to predict the future: let Bob have an ability to predict Alice’s will. Now, Bob prepares two boxes, Box1 and Box2, and Alice can select either Box2 or both boxes. Box1 contains $1. Box2 contains $1,000 only if Alice selects only Box2; otherwise Box2 is empty($0). Which is better for Alice? Since Alice cannot decide which one is better in general, this problem is called Newcomb’s paradox. In this paper, we propose quantum strategies for this paradox by Bob having quantum ability. Many other results including quantum strategies put emphasis on finding out equilibrium points. On the other hand, our results put emphasis on whether a player can predict another player’s will. Then, we show some positive solutions for this problem.

DOI
10.4236/ib.2010.21004

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Newcomb’s problem is a game between two players, one of who has an ability to predict the future: let Bob have an ability to predict Alice’s will. Now, Bob prepares two boxes, Box1 and Box2, and Alice can select either Box2 or both boxes. Box1 contains 1.Box2contains1. Box2 contains 1.Box2contains1,000 only if Alice selects only Box2; otherwise Box2 is empty($0). Which is better for Alice? Since Alice cannot decide which one is better in general, this problem is called Newcomb’s paradox. In this paper, we propose quantum strategies for this paradox by Bob having quantum ability. Many other results including quantum strategies put emphasis on finding out equilibrium points. On the other hand, our results put emphasis on whether a player can predict another player’s will. Then, we show some positive solutions for this problem.
In a recently introduced coset guessing game, Alice plays against Bob and Charlie, aiming to meet a joint winning condition. Bob and Charlie can only communicate before the game starts to devise a joint strategy. The game we consider begins with Alice preparing a 2m-qubit quantum state based on a random selection of three parameters. She sends the first m qubits to Bob and the rest to Charlie, and then reveals to them her choice for one of the parameters. Bob is supposed to guess one of the hidden parameters, Charlie the other, and they win if both guesses are correct. From previous work, we know that the probability of Bob’s and Charlie’s guesses being simultaneously correct goes to zero exponentially as m increases. We derive a tight upper bound on this probability and show how Bob and Charlie can achieve it. While developing an optimal strategy, we devised an encoding circuit using only and Hadamard gates, which builds CSS codes from EPR pairs using only local operations. We found that the role of quantum information that Alice communicates to Bob and Charlie is to make their responses correlated rather than improve their individual (marginal) correct guessing rates.
We present a guessing to be called secret-bit guessing game, where a player (say, Bob) guesses two secret bits enveloped by another player (say, Alice) based on a legitimate strategy. Here we introduce a notion of supplementary information, under the name of hint, available for Bob to guess the secret bits. In such setting, we investigate an intriguing question: What happens if the physical elements to realize Bob's strategy are quantum? On the basis of the analysis of payoff function, we here offer a new scenario, namely, of allowing Bob to make better use of a given hint with the quantum elements beyond their classical counterparts. We argue that this quantum advantage is possible due to the ability of the quantum elements to use the quality of a given hint, as well as its amount, faithfully dealing with the nature of quantum superposition.
We revisit the quantum two-person duel. In this problem, both Alice and Bob each possess a spin-1/2 particle which models dead and alive states for each player. We review the Abbott and Flitney result—now considering non-zero α1 and α2 in order to decide if it is better for Alice to shoot or not the second time—and we also consider a duel where players do not necessarily start alive. This simple assumption allows us to explore several interesting special cases, namely how a dead player can win the duel shooting just once, or how can Bob revive Alice after one shot, and the better strategy for Alice—being either alive or in a superposition of alive and dead states—fighting a dead opponent.