Clues About Bluffing in Clue: Is Conventional Wisdom Wise?

Digital Commons @ George Fox University Faculty Publications - Department of Electrical Department of Electrical Engineering and Engineering and Computer Science Computer Science 2019 Clues About Bluffing in Clue: Is Conventional Wisdom Wise? David Hansen Kyle D. Hansen Follow this and additional works at: https://digitalcommons.georgefox.edu/eecs_fac Part of the Engineering Commons Clues About Bluffing in Clue : Is Conventional Wisdom Wise? David M. Hansen Affiliate Member, IEEE1, Kyle D. Hansen2 1College of Engineering, George Fox University, Newberg, OR, USA 2Westmont College, Santa Barabara, CA, USA We have used the board game Clue as a pedagogical tool in our course on Artificial Intelligence to teach formal logic through the development of logic-based computational game-playing agents. The development of game-playing agents allows us to experimentally test many game-play strategies and we have encountered some surprising results that refine “conventional wisdom” for playing Clue. In this paper we consider the effect of the oft-used strategy wherein a player uses their own cards when making suggestions (i.e., “bluffing”) early in the game to mislead other players or to focus on acquiring a particular kind of knowledge. We begin with an intuitive argument against this strategy together with a quantitative probabilistic analysis of this strategy’s cost to a player that both suggest “bluffing” should be detrimental to winning the game. We then present our counter-intuitive simulation results from playing computational agents that “bluff” against those that do not that show “bluffing” to be beneficial. We conclude with a nuanced assessment of the cost and benefit of “bluffing” in Clue that shows the strategy, when used correctly, to be beneficial and, when used incorrectly, to be detrimental. Index Terms—Benchmarking, Board games, Competitions, Multi-player games, Strategy games I. INTRODUCTION In our AI course, students develop “agents” (i.e., computer programs) that are capable of playing Clue using a game- For many years we have used Clue R 1 as a pedagogical tool2 server we have developed that enables students to compete in our Artificial Intelligence course at George Fox University against one another, and faculty, to test the logic and strategies as a motivating platform for the development of “intelligent” employed by their agents. Over time, some interesting results computational agents [1]. have emerged with respect to the strategies that are most Clue is a strategy game of reasoning with limited knowledge effective at winning the game. Somewhat surprisingly, one to deduce an initially hidden solution — the “who, what strategy many human players and student agents routinely where” of a murder. The game’s domain consists of 6 persons employ — using cards in a suggestion that they hold in their (“suspects”), 6 weapons, and 9 rooms, each represented by hand — is not well-studied yet assumed to be beneficial.6 a card in the physical game. Game setup involves drawing We begin with a discussion of common logical approaches one random card from each category (suspect, weapon, room) to playing the game of Clue. We then present an intuitive and and hiding them as the solution. The remaining 18 cards are probabilistic analysis of the effects of using one’s own cards shuffled and dealt to the players.3 in suggestions followed by counter-intuitive experimental ev- The game is played on a physical board that depicts the idence of the effect of using this strategy. rooms distributed in space and, on each turn, a player rolls a die to move about the board with the intent of entering a room II. RELATED WORK where they can make a guess about what suspect, weapon, and room are in the solution. These guesses as referred to as Clue has sometimes been used as a pedagogical device “suggestions” and the one restriction on a suggestion is that for teaching formal logic [2], [3]. Others have used Clue to it must include the room where the player is located. When a investigate how to develop higher-level systems for machine player makes a suggestion, opponents are polled in turn-order learning and automated theorem proving [4]–[6]. While these as to whether or not they are able to “refute” the suggestion approaches formalize the logic for playing a legal and even by privately revealing to the suggester alone which card in the “intelligent” game of Clue, they do not address game-play suggestion the opponent holds. Once the suggestion has been strategy in general, nor the use of one’s own cards in sugges- refuted, no further opponents are polled and the turn passes tions in particular. to the next player. Once a player has deduced the hidden set However, non-academic web sites with discussions of strate- of cards in the solution they may make a final game-winning gies for playing Clue endorse using one’s own cards in “accusation” on their turn.45 suggestions with a variety of rationales that generally include 1) confusing opponents or 2), gathering information on a 1Clue is known outside the United States as Cluedo R . particular kind of card [7]–[10]. 2Calling it Glomus — a rough Latin translation of the word “clue” being With respect to the strategy of using cards one holds, a skein of yarn of the sort used by Ariadne to escape the maze in the myth of the Minotaur. Thomas Ferguson addresses what he denotes as “bluffing” in 3Games can be played by 2-6 players and cards in games of 4 or 5 players the context of “Two-Person Zero-Sum Games” in Part II of will not be dealt evenly. 4There is restriction that the player occupy the room used in their accusation 6Students in our course who are familiar with the game routinely discuss 5A false accusation ends the game for a mistaken player. and use this strategy. his textbook on Game Theory [11]. Ferguson describes a Clue- in some player’s hand. We might express this deductive rule like game “Guess It!” to demonstrate the need for “bluffing” concisely using predicate logic by a player to prevent an opponent from winning a game by isInSolution(Card) presuming the player is always “honest”: From a deck with m + n + 1 distinct cards, 9!Card8P layer(:has(P layer; Card)) m cards are dealt to Player I, n cards are dealt to A player can be known to have a card in their hand if Player II, and the remaining card, called the “target • I am the player and have the card in my hand, or card”, is placed face down on the table. Players know • a player refuted my suggestion by showing the card to their own cards but not those of their opponent. me, or The objective is to guess correctly the target card. • a player refuted a suggestion and we know that the player Players alternate moves, with Player I starting. At can not have in their hand the other two cards used in each move, a player may either the suggestion (1) guess at the target card, in which case the Most casual board game players rely on the first two mecha- game ends, with the winner being the player who nisms and omit the third that depends on determining what guessed if the guess is correct, and his opponent if cards other players can not have in their hands. Others the guess is incorrect, or formally [2], [3] and informally [10], [13], [14] observe that, (2) ask if the other player holds a certain card. with some additional note-taking, the “process of deduction” If the other player has the card, that card must be can also be employed by knowing what cards players do not shown and is removed from play. have in their hand — a card all players can not have therefore With a deck of say 11 cards and each player being part of the solution. Leveraging a player’s inability to receiving 5 cards, this is a nice playable game that refute a suggestion leads to the second proof technique. illustrates need for bluffing in a clear way. If a player The second proof technique is through the “process of asks about a card that is in his own hand, he knows deduction.” This technique relies on deducing and inferring what the answer will be. We call such a play a bluff. what cards are in the solution by observing what cards players If a player asks about a card not in his hand, we say can not have in their hands and drawing an indirect conclusion he is honest. If a player is always honest and the card about what card is part of the solution. Thus a card is known he asks about is the target card, the other player will to be part of the solution when it is known that every player know that the requested card is the target card and can not have it in their hand so will win. Thus a player must bluff occasionally. Bluffing may also lure the opponent into a wrong isInSolution(Card) guess at the target card [11, p II-63]. 8P layer(canNotHave(P layer; Card)) While useful, this limits consideration of “bluffing” and “honesty” to the context of Two-Person Zero-Sum Games such A player can be known to not have a card if as “Guess-It!”; these strategies are not addressed in multi- • the player was unable to refute a suggestion that included player games such as Clue. Moreover, Ferguson does not the card, or address the strategic question of when or how often one should • some other player is known to have the card, or bluff to optimize one’s chance of winning, perhaps due to the • the card is known to be part of the solution difficulty of integrating “higher-order information (what are An observant reader may notice the “indirect recursive” the other players thinking)” [12].

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