A Statistical View of Learning in the Centipede Game

In this article we evaluate the statistical evidence that a population of students learn about the sub-game perfect Nash equilibrium of the centipede game via repeated play of the game. This is done by formulating a model in which a player's error in assessing the utility of decisions changes as they gain experience with the game. We first estimate parameters in a statistical model where the probabilities of choices of the players are given by a Quantal Response Equilibrium (QRE) (McKelvey and Palfrey, 1995, 1996, 1998), but are allowed to change with repeated play. This model gives a better fit to the data than similar models previously considered. However, substantial correlation of outcomes of games having a common player suggests that a statistical model that captures within-subject correlation is more appropriate. Thus we then estimate parameters in a model which allows for within-player correlation of decisions and rates of learning. Through out the paper we also consider and compare the use of randomization tests and posterior predictive tests in the context of exploratory and confirmatory data analyses

Nerve injury and neuropathic pain - A question of age

The effects of peripheral nerve injury on somatosensory processing and pain are highly dependent upon the age at which the damage occurs. Adult nerve injury rapidly triggers neuropathic pain, but this is not so if the same nerve injury is performed in animals below postnatal day (P) 28, consistent with observations in paediatric patients. However, longitudinal studies show that pain hypersensitivity emerges later in life, when the animal reaches adolescence, an observation that could be of clinical importance. Here we discuss the evidence that the central consequences of nerve damage are critically determined by the status of neuroimmune regulation at different ages. In the first postnatal weeks, when spinal somatosensory circuits are undergoing synaptic reorganisation, the 'default' neuroimmune response is skewed in an anti-inflammatory direction, suppressing the excitation of dorsal horn neurons and preventing the onset of neuropathic pain. As animals grow up and the central nervous system matures, the neuroimmune profile shifts in a pro-inflammatory direction, unmasking a 'latent' pain response to an earlier nerve injury. The data predicts that nerve injury in infancy and childhood could go unnoticed at the time, but emerge as clinically 'unexplained' or 'functional' pain in adolescence

Nanoscale electrochemical patterning reveals the active sites for catechol oxidation at graphite surfaces

Graphite-based electrodes (graphite, graphene, and nanotubes) are used widely in electrochemistry, and there is a long-standing view that graphite step edges are needed to catalyze many reactions, with the basal surface considered to be inert. In the present work, this model was tested directly for the first time using scanning electrochemical cell microscopy reactive patterning and shown to be incorrect. For the electro-oxidation of dopamine as a model process, the reaction rate was measured at high spatial resolution across a surface of highly oriented pyrolytic graphite. Oxidation products left behind in a pattern defined by the scanned electrochemical cell served as surface-site markers, allowing the electrochemical activity to be correlated directly with the graphite structure on the nanoscale. This process produced tens of thousands of electrochemical measurements at different locations across the basal surface, unambiguously revealing it to be highly electrochemically active, with step edges providing no enhanced activity. This new model of graphite electrodes has significant implications for the design of carbon-based biosensors, and the results are additionally important for understanding electrochemical processes on related sp2-hybridized materials such as pristine graphene and nanotubes

Engodeneity of Alternating Offers in a Bargaining Game

We investigate an infinite horizon two-person simultaneous offer bargaining game of incomplete information with discounted playoffs. In each period, each player chooses to give in or hold out. The game continues until at least one of the players chooses to give in, at which point agreement has been reached and the game terminates, with an agreement benefit accruing to each player, and a cost to the player (or players) that give in. Players have privately known agreement benefits. 'Low benefit players have a weakly dominant strategy to hold out forever; high benefit players would be better off giving in if they knew their opponent was planning to hold out forever. For any discount factor there is a unique Nash equilibrium in which the two players alternate in their willingness to give in, if the players' priors about each others type are sufficiently asymmetric. Second, for almost all priors, this is the unique equilibrium if the discount factor is close enough to one

An Experimental Study of the Centipede Game

We report on an experiment in which individuals play a version of the centipede game. In this game, two players alternately get a chance to take the larger portion of a continually escalating pile of money. As soon as one person takes, the game ends with that player getting the larger portion of the pile, and the other player getting the smaller portion. If one views the experiment as a complete information game, all standard game theoretic equilibrium concepts predict the first mover should take the large pile on the first round. The experimental results show that this does not occur. An alternative explanation for the data can be given if we reconsider the game as a game of incomplete information in which there is some uncertainty over the payoff functions of the players. In particular, if the subjects believe there is some small likelihood that the opponent is an altruist, then in the equilibrium of this incomplete information game, players adopt mixed strategies in the early rounds of the experiment, with the probability of taking increasing as the pile gets larger. We investigate how well a version of this model explains the data observed in the centipede experiments

Quantal Response Equilibria for Extensive Form Games

This paper investigates the use of standard econometric models for quantal choice to study equilibria of extensive form games. Players make choices based on a quantal choice model, and assume other players do so as well. We define an Agent Quantal Response Equilibrium (AQRE), which applies QRE to the agent normal form of an extensive form game and imposes a statistical version of sequential rationality. We also define a parametric specification, called logit-AQRE, in which quantal choice probabilities are given by logit response functions. AQRE makes predictions that contradict the invariance principle in systematic ways. We show that these predictions match up with some experimental findings by Schotter, Weigelt and Wilson (1993) about the play of games that differ only with respect to inessential transformations of the extensive form. The logit-AQRE also implies a unique selection from the set of subgame perfect equilibria in generic extensive form games. We examine data from signalling game experiments by Banks, Camerer, and Porter (1994) and Brandts and Holt (1993). We find that the logit-AQRE selection applied to these games succeeds in predicting patterns of behavior observed in these experiments, even when our prediction conflicts with more standard equilibrium refinements, such as the intuitive criterion. We also reexamine data from the McKelvey and Palfrey (1992) centipede experiment

The Holdout Game: An Experimental Study of an Infinitely Repeated Game with Two-Sided Incomplete Information

We investigate experimentally a two-person infinitely repeated game of incomplete information. In the stage game, each player chooses to give in or hold out. Players have privately known costs of giving in and each player receives a fixed benefit whenever at least one player gives in. High cost players have a dominant strategy in the stage game to hold out, and the low cost players ' best response depends on what the opponent does. Equilibrium play to the infinitely repeated game conveys information about the players’ type. We investigate two questions: whether there is any evidence that subject behavior approximates belief stationary equilibria, and whether there is evidence that subjects will converge to an equilibrium of the correct state. We conclude that subjects do not adopt symmetric belief stationary strategies for the holdout game. However, we cannot reject the hypotheses that subjects converge towards eventually playing an equilibrium of the correct state (even though they do not always learn the correct state). Behavior of experienced subjects is closer to the predictions of symmetric belief-stationary equilibrium

Performance of charge-injection-device infrared detector arrays at low and moderate backgrounds

Three 2 x 64 element charge injection device infrared detector arrays were tested at low and moderate background to evaluate their usefulness for space based astronomical observations. Testing was conducted both in the laboratory and in ground based telescope observations. The devices showed an average readout noise level below 200 equivalent electrons, a peak responsivity of 4 A/W, and a noise equivalent power of 3x10 sq root of W/Hz. Array well capacity was measured to be significantly smaller than predicted. The measured sensitivity, which compares well with that of nonintegrating discrete extrinsic silicon photoconductors, shows these arrays to be useful for certain astronomical observations. However, the measured readout efficiency and frequency response represent serious limitations in low background applications

Nanoscale intermittent contact-scanning electrochemical microscopy

A major theme in scanning electrochemical microscopy (SECM) is a methodology for nanoscale imaging with distance control and positional feedback of the tip. We report the expansion of intermittent contact (IC)-SECM to the nanoscale, using disk-type Pt nanoelectrodes prepared using the laser-puller sealing method. The Pt was exposed using a focused ion beam milling procedure to cut the end of the electrode to a well-defined glass sheath radius, which could also be used to reshape the tips to reduce the size of the glass sheath. This produced nanoelectrodes that were slightly recessed, which was optimal for IC-SECM on the nanoscale, as it served to protect the active part of the tip. A combination of finite element method simulations, steady-state voltammetry and scanning electron microscopy for the measurement of critical dimensions, was used to estimate Pt recession depth. With this knowledge, the tip-substrate alignment could be further estimated by tip approach curve measurements. IC-SECM has been implemented by using a piezo-bender actuator for the detection of damping of the oscillation amplitude of the tip, when IC occurs, which was used as a tip-position feedback mechanism. The piezo-bender actuator improves significantly on the performance of our previous setup for IC-SECM, as the force acting on the sample due to the tip is greatly reduced, allowing studies with more delicate tips. The capability of IC-SECM is illustrated with studies of a model electrode (metal/glass) substrate