Probability estimation in poker: A qualified success for unaided judgment

Poker players make strategic decisions on the basis of imperfect information, which are informed by their assessment of the probability they will hold the best set of cards among all players at the conclusion of the hand. Exact mental calculations of this probability are impossible—therefore, players must use judgment to estimate their chances. In three studies, 69 moderately experienced poker players estimated the probability of obtaining the best cards among all players, based on the limited information that is known in the early stages of a hand. Although several of the conditions typically associated with well‐calibrated judgment did not apply, players' judgments were generally accurate. The correlation between judged and true probabilities was r > .8 for over five‐sixths of the participants, and when judgments were averaged across players and within hands this correlation was .96. Players slightly overestimated their chance of obtaining the best cards, mainly where this probability was low to moderate (<.7). Probability estimates were slightly too strongly related to the strength of the two cards that a player holds (known only to themselves), and insufficiently influenced by the number of opponents. Seemingly, players show somewhat insufficient regard for the cards that other players could be holding and the potential for opponents to acquire a strong hand. The results show that even when judgment heuristics are used to good effect in a complex probability estimation task, predictable errors can still be observed at the margins of performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Synergistic Information Processing Encrypts Strategic Reasoning in Poker

There is a tendency in decision‐making research to treat uncertainty only as a problem to be overcome. But it is also a feature that can be leveraged, particularly in social interaction. Comparing the behavior of profitable and unprofitable poker players, we reveal a strategic use of information processing that keeps decision makers unpredictable. To win at poker, a player must exploit public signals from others. But using public inputs makes it easier for an observer to reconstruct that player's strategy and predict his or her behavior. How should players trade off between exploiting profitable opportunities and remaining unexploitable themselves? Using a recent multivariate approach to information theoretic data analysis and 1.75 million hands of online two‐player No‐Limit Texas Hold'em, we find that the important difference between winning and losing players is not in the amount of information they process, but how they process it. In particular, winning players are better at integrative information processing—creating new information from the interaction between their cards and their opponents’ signals. We argue that integrative information processing does not just produce better decisions, it makes decision‐making harder for others to reverse engineer, as an expert poker player's cards act like the private key in public‐key cryptography. Poker players encrypt their reasoning with the way they process information. The encryption function of integrative information processing makes it possible for players to exploit others while remaining unexploitable. By recognizing the act of information processing as a strategic behavior in its own right, we offer a detailed account of how experts use endemic uncertainty to conceal their intentions in high‐stakes competitive environments, and we highlight new opportunities between cognitive science, information theory, and game theory.     

Manipulating Games by Sharing Information

We address the issue of manipulating games through communication. In the specific setting we consider (a variation of Boolean games), we assume there is some set of environment variables, the values of which are not directly accessible to players; the players have their own beliefs about these variables, and make decisions about what actions to perform based on these beliefs. The communication we consider takes the form of (truthful) announcements about the values of some environment variables; the effect of an announcement is the modification of the beliefs of the players who hear the announcement so that they accurately reflect the values of the announced variables. By choosing announcements appropriately, it is possible to perturb the game away from certain outcomes and towards others. We specifically focus on the issue of stabilisation: making announcements that transform a game from having no stable states to one that has stable configurations.     

Logics for Epistemic Programs

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents' uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents' uncertainty about the current state of the system. Program models induce changes affecting agents' information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model. Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof. The basic operation used in the semantics is called the update product. A version of this was introduced in Baltag et al. (1998), and the presentation here improves on the earlier one. The update product is used to obtain from any program model the corresponding epistemic update, thus allowing us to compute changes of information or belief. This point is of interest independently of our logical languages. We illustrate the update product and our logical languages with many examples throughout the paper.     

An n-Player Semantic Game for an n + 1-Valued Logic

First we show that the classical two-player semantic game actually corresponds to a three-valued logic. Then we generalize this result and give an n-player semantic game for an n + 1-valued logic with n binary connectives, each associated with a player. We prove that player i has a winning strategy in game if and only if the truth value of is t _i in the model M, for 1 ≤ i ≤ n; and none of the players has a winning strategy in if and only if the truth value of is t ₀ in M.     

Information and the complementarity game

Complementarity games are explorative games between two players. The deterministic rules of the game are known to both players. The first player freely chooses among a finite number of states of the game but tells nobody of this decision. The second player accesses the game only by primitive, symbolic input‐output interfaces. The task of the second player is to find out the unknown state by input‐output experiments. The logic resulting from this scenario resembles‐ quantum logic in many aspects. It is the empirical logic of choice for virtual realities.     

Keep ‘hoping’ for rationality: a solution to the backward induction paradox

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.     

Inference and update

We look at two fundamental logical processes, often intertwined in planning and problem solving: inference and update. Inference is an internal process with which we uncover what is implicit in the information we already have. Update, on the other hand, is produced by external communication, usually in the form of announcements and in general in the form of observations, giving us information that might not have been available (even implicitly) before. Both processes have received attention from the logic community, usually separately. In this work, we develop a logical language that allows us to describe them together. We present syntax, semantics and a complete axiom system; we discuss similarities and differences with other approaches and mention how the work can be extended.     

Socially Constructed Technology

The concepts of omniscience and omnipotence are defined in 2 × 2 ordinal games, and implications for the optimal play of these games, when one player is omniscient or omnipotent and the other player is aware of his omniscience or omnipotence, are derived. Intuitively, omniscience allows a player to predict the strategy choice of an opponent in advance of play, and omnipotence allows a player, after initial strategy choices are made, to continue to move after the other player is forced to stop. Omniscience and its awareness by an opponent may hurt both players, but this problem can always be rectified if the other player is omniscient. This pathology can also be rectified if at least one of the two players is omnipotent, which can override the effects of omniscience. In some games, one player's omnipotence ‐ versus the other's ‐ helps him, whereas in other games the outcome induced does not depend on which player is omnipotent. Deducing whether a player is superior (omniscient or omnipotent) from the nature of his game playing alone raises several problems, however, suggesting the difficulty of devising tests for detecting superior ability in games.     

TRIANGLE EFFECT AND THE CONNOTATIVE MEANING OF TRUST IN PRISONER'S DILEMMA: A CROSS CULTURAL STUDY

Two hundred and forty-nine male postgraduate students of management played the Prisoner's Dilemma Game (Deutsch 1960) and filled out a postgame questionnaire measuring attitude toward the ‘other player’. Striking differences resulted between trusting and trustworthy subjects on the one hand and suspicious and untrustworthy subjects on the other with respect to different meanings given to the dimension of trust (cooperation) in the interaction. As predicted, trusting behavior of the other player was given a positive evaluative meaning - good versus bad - by the trusting and trustworthy subjects and negative dynamism meaning - weak versus strong - by the suspicious and untrustworthy subjects. The trusting players expected the typical other to make either trusting or suspicious moves, whereas the suspicious subjects expected the typical other to be uniformly suspicious, yielding a high Triangularity Index (Kelley and Stahelski 1970). Most provocatively, while 51% of trusting subjects thought that the other player was a female, 81% from among the suspicious subjects thought so. Some implications of the results in interpersonal and organizational situations are discussed.     

The hot hand exists in volleyball and is used for allocation decisions

The "hot hand" belief in sports refers to the conviction that a player has a higher chance of making a shot after two or three successful shots than after two or three misses (resulting in "streaks"). This belief is usually considered a cognitive fallacy, although it has been conjectured that in basketball the defense will attack a "hot" player and prevent streaks from occurring. To address this argument, we provide the first study on the hot hand in volleyball, where the net limits direct defensive counterstrategies, meaning that streaks can more likely emerge if a player is hot. We first establish that athletes believe in the hot hand in volleyball (Study 1A). Analyzing the top 26 first-division players, we then show that streaks do exist for half of the players (Study 1B). Coaches can detect players' performance variability and use it to make strategic decisions (Study 2A). Playmakers are also sensitive to streaks and rely on them when deciding to whom to allocate the ball (Study 2B). We conclude that for volleyball the hot hand exists, coaches and playmakers are able to detect it, and playmakers tend to use it "adaptively," which results in more hits for a team.     

Public and private communication are different: results on relative expressivity

Dynamic Epistemic Logic (DEL) is the study of how to reason about knowledge, belief, and communication. This paper studies the relative expressivity of certain fragments of the DEL language for public and private communication. It is shown that the language of public communication with common knowledge and the language of private communication with common knowledge are expressively incomparable for the class of all pointed Kripke models, which provides a formal proof that public and private communication are fundamentally different in the presence of common knowledge. It is also shown that single-recipient private communication does not add expressive power to the language of modal logic with common knowledge for any class of transitive pointed Kripke models. The latter result provides a sense in which positive introspection—believing our own beliefs—induces a kind of self-dialog.     

Games with Zero-knowledge Signaling

We observe that in certain two-player repeated games of incomplete information, where information may be incomplete on both sides, it is possible for an informed player to signal his status as an informed player to the other without revealing any information about the choice of chance. The key to obtaining such a class of games is to relax the assumption that the players’ moves are observable. We show that in such cases players can achieve a kind of signaling that is “zero-knowledge”, in the sense that the other player becomes convinced that her opponent is informed without ever learning the choice of chance. Moreover, such “zero-knowledge signaling” has all of the statistical properties associated with zero-knowledge proofs in intereactive protocols. In particular, under the general assumption that moves are unobservable, such signaling leads to a class of equilibria in repeated games that are separatingin regard to the status of player 1–informed or uninformed–but only for player 2; any other player in a network, being unable to observe the moves of player 2, remains uncertain as to the status of player 1. Special Issue Formal Epistemology II. Edited by Branden Fitelson     

Extending probabilistic dynamic epistemic logic

This paper aims to extend in two directions the probabilistic dynamic epistemic logic provided in Kooi’s paper (J Logic Lang Inform 12(4):381–408, 2003) and to relate these extensions to ones made in van Benthem et al. (Proceedings of LOFT’06. Liverpool, 2006). Kooi’s probabilistic dynamic epistemic logic adds to probabilistic epistemic logic sentences that express consequences of public announcements. The paper (van Benthem et al., Proceedings of LOFT’06. Liverpool, 2006) extends (Kooi, J Logic Lang Inform 12(4):381–408, 2003) to using action models, but in both papers, the probabilities are discrete, and are defined on trivial σ-algebras over finite sample spaces. The first extension offered in this paper is to add a previous-time operator to a probabilistic dynamic epistemic logic similar to Kooi’s in (J Logic Lang Inform 12(4):381–408, 2003). The other is to involve non-trivial σ-algebras and continuous probabilities in probabilistic dynamic epistemic logic.     

Toward an Understanding of Players’ Perceptions of Talent Development Environments in UK Female Football

This study explored UK female football players' perceptions of their talent development environments using the Talent Development Environment Questionnaire. Participants were 137 UK-based female football players (M_age = 16.06, SD = 1.90) from Football Association Girls' Centres of Excellence and Football Association Women's Super League Development Squads. Players had most positive perceptions of long-term development focus and support network, whereas the least positive perceptions were of communication and understanding the athlete. Sport psychologists could offer significant support in (a) planning for football-specific development and career progression, (b) communication with key social agents, and (c) holistic player development and well-being.     

Get angry, get out: The interpersonal effects of anger communication in multiparty negotiation

Research on multiparty negotiation has investigated how parties form coalitions to secure payoffs but has not addressed how emotions may affect such coalition decisions. Extending research on bilateral negotiations which has generally argued that it is beneficial to communicate anger, we argue that it constitutes a considerable risk when there are more than two people present at the negotiation table. Using a computer-mediated coalition game we show that communicating anger is a risky strategy in multiparty bargaining. The main findings of three studies were that participants: (1) form negative impressions of players who communicate anger and therefore (2) exclude such players from coalitions and from obtaining a payoff share, but (3) make considerable concessions on those rare occasions that they choose to form a coalition with an angry player, or (4) when they had to form a coalition with an angry player. We discuss the implications of these results for theorizing on emotions, negotiations, and coalition formation.     

Subgames within Large Games and the Heuristic of Imitation

We study repeated normal form games where the number of players is large. We argue that it is interesting to look at such games as being divided into subgames, each of which we call a neighbourhood. The structure of such a game is given by a graph G whose nodes are players and edges denote visibility. The neighbourhoods are maximal cliques in G. The game proceeds in rounds where in each round the players of every clique X of G play a strategic form game among each other. A player at a node v strategises based on what she can observe, i.e., the strategies and the outcomes in the previous round of the players at vertices adjacent to v. Based on this, the player may switch strategies in the same neighbourhood, or migrate to another neighbourhood. We are interested in addressing questions regarding the eventual stability of such games. We incrementally impose constraints on the ‘types’ of the players. First, we look at players who are unconstrained in their strategising abilities, in that, players who may use unbounded memory. We then consider the case of memoryless players. We show that in both these cases the eventual stability of the game can be characterised in terms of potentials. We then introduce a simple modal logic in which the types of the players can be specified. We show that when the players play according to these specified types, it can be effectively decided whether the game stabilises. Finally, we look at the important heuristic of imitation. Simple imitative strategies can be specified in the logic introduced by us. We show that in a population of optimisers and imitators, we can decide how ‘worse-off’ the imitators are by playing imitative strategies rather than optimal ones.     

Optimism and Positivity Biases in Performance Appraisal Ratings: Empirical Evidence from Professional Soccer

Using unique assessment data for players from a German Bundesliga club’s youth academy, we tested four core hypotheses on how player ratings and rater or ratee-related characteristics reflect the (prospective) optimism bias and (retrospective) positivity bias. The results indicate not only that the ratings of predicted and remembered performance are indeed higher than the talents’ actual performance throughout a season, but that these differences depend positively on the rater’s organizational experience and negatively on the amount of ratee data available. They also suggest that (prospective) anticipation is even more positively biased than (retrospective) recollection of player performances, underscoring the asymmetry between looking forward and looking backward.     

Circle graphs and monadic second-order logic

This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic second-order formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.