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With a mixture of ambitious, financially backed clubs looking to hit the big time, former Premier League giants that have fallen on hard times and ufc betting odds 15206 lower-league clubs desperate to avoid the drop into non-league, there is so much riding on the English fourth tier each and every season. Salford City, backed by former Manchester United superstars such as Gary Neville and David Beckham, will compete with another north-west club in the shape of Bolton Wanderers fixtures football league 2 betting if stoke city vs liverpool betting expert boxing fixtures football league 2 betting are to be believed. There will be shocks, surprises, thrills and spills along the way and you can count on the Squawka Bet experts to put in the time it takes to research and analyse the League Two betting markets to bring you our best predictions and tips every single step of the way. As with all of our tips, we do our utmost to get our predictions live and ready for you at least 48 hours before kick-off time. We pride ourselves on our depth of knowledge of English football too, right down to League Two and even beyond that, plus our extensive access to the data helps us to select the best bets for each round of fixtures. There are plenty of opportunities for League Two inplay betting nowadays. Whilst placing a mid-match bet used to be very difficult and in some cases limited only to certain matches, it now appears that the vast majority of games across the world are available to bet on as the action unfolds and the English fourth tier is in no way exempt from this, with live, in-play odds available on almost every single Football League match across a game campaign.

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Betting odds explained 10/325

Those are players who have four accrued seasons, and are free to sign with You'll now receive the top Texans Wire stories each day directly in your inbox. Please enter an email address. Something went wrong. October 15, The former offensive lineman for the New Orleans Saints and Green From The Web Ads by Zergnet. Houston Texans are 3rd in waiver wire priority Mark Lane. Yates commented, "Good opportunity for Jacksonville to have the first chance at young talent Here are the three types of free agents the Texans have.

Share this article shares share. Most Popular. Email Sign up No, thanks. Never miss a great story Start every day with our most popular content waiting in your inbox. An error has occured Please re-enter your email address. Thanks for signing up! Error Please enter an email address. Indeed, a number of researchers have recently attempted to characterize the processes by which participants use outcome feedback to search for an optimal strategy in the binary prediction task e.

If the impact of feedback was to spur the generation of alternative strategies, which eventually led to identification of maximizing as the optimal response, then we should see an interaction whereby little or no effect of feedback would be found among those participants given the hint for whom the maximizing strategy was already highly available. This should be especially true of the subset of participants who, when prompted with that hint, correctly identified the maximizing strategy as superior.

An alternative possibility was that the hint effect observed by Koehler and James would not generalize to tasks involving trial-by-trial predictions with feedback. In their studies, the participants only made predictions for ten rolls, and did so without any information about the outcomes. In a feedback version, it is possible that seeing the regular occurrence of the low-probability outcome might make it difficult to stick to the maximizing strategy, even if, when prompted, one could correctly identify it as optimal cf.

Goodnow, The contribution of our new experiments was that they allowed us to examine the joint influences of two manipulations—feedback and provision of a hint—that up to now have only been studied in isolation. The principal differences between Experiments 1 and 2 were the use of a real Exp. The real die was used in Experiment 1 to counter the criticism that participants might make suboptimal predictions in virtual tasks because they doubt that the die is fair and that the probabilities are stationary e.

In Experiment 1 , participants saw the same physical die being rolled on every trial, thus presumably eliminating, or at least substantially reducing, any skepticism about the task parameters. A total of undergraduate students from the University of Waterloo participated in return for course credit and earnings from the experiment.

They were randomly assigned to one of the four experimental conditions with n per condition varying from 29 to The data from eight additional participants were excluded from the analysis: five who had completed a similar task before, two who failed to follow the instructions, and one who was red—green colorblind. Participants predicted the outcomes for a series of 50 rolls of a ten-sided die with seven green sides and three red sides.

The experimenter, who was present when the predictions were made, rolled the die to determine the outcomes. This was done either after each prediction feedback condition or after all 50 predictions had been made no feedback condition. The outcomes were recorded directly on the prediction sheet, and the total winnings were calculated and paid at the end of the session.

Before completing the prediction task, one group of participants hint condition read a description of two possible strategies that could be used in the game. Which strategy do you think will win more money? Footnote 2. Error bars indicate SEM s. There was a slight tendency for predicting the dominant color more often in the feedback than in the no-feedback condition M of 8. The proportions choosing maximizing were virtually identical in the hint no-feedback The upper panels of Fig.

The figures show that not all participants followed the strategy that they had just endorsed when making their trial-by-trial predictions. However, a comparison of the upper two hint and lower two no hint panels shows that a much higher percentage of participants endorsed maximizing in the hint condition black bars, upper panels than spontaneously chose green for all ten rolls of the first game in the no-hint condition far right open bars, lower panels.

This pattern is consistent with the idea that the hint prompts participants to realize that maximizing is optimal. The mean number of green predictions across all trials for participants who identified maximizing matching as superior was 8. The key finding of Experiment 1 is that providing a hint designed to make the maximizing and matching strategies equally available increased levels of maximizing on the subsequent prediction task.

In contrast, provision of feedback about the outcome of each die roll did not significantly increase maximizing. One difference, however, between Experiment 1 and the previous research was the relatively small number of prediction trials. Newell and Rakow found that significant increases in the rates of maximization only occurred after or more prediction trials.

This allowed us not only to attempt to replicate the findings of Experiment 1 and of Newell and Rakow, but also to examine the interplay of the hint and feedback over an extended number of predictions. The increased number of training trials also facilitated the application of reinforcement learning models to our data. These were applied in an effort to find converging evidence of the roles played by the hint and feedback in the binary prediction task. In the two feedback-present conditions, an image of a ten-sided die rolled across the screen following each prediction and the outcome was displayed.

In the two feedback-absent conditions, no die was shown and the participants were told that the die would be rolled once all of the predictions had been made, in order to determine payment. The dominant color i. Mean proportions of dominant-color guesses for each of the six blocks of 50 rolls in Experiment 2. To provide a like-for-like comparison between Experiments 1 and 2 , we analyzed the initial 50 trials of Experiment 2 i.

This replicates the results of Experiment 1 for the equivalent number of trials. It shows that the change from a real to a virtual die, the change in population Canadian vs. Australian participants , and the absence of the experimenter in Experiment 2 did not significantly alter the results. Neither of these individual block analyses showed significant interactions between hint and feedback. In short, the hint manipulation had an influence on early prediction trials but not on later ones, leading to no overall main effect of hint.

This effect is especially prominent in the no-hint no-feedback group, in which matching was overwhelmingly dominant in the first ten-roll game lower right panel. Numbers of dominant-color guesses made by participants in the first ten-roll game of Experiment 2. To investigate in more detail the interplay of feedback and hint over the entire trials, we examined the prediction data across blocks for the subset of participants in the hint conditions who explicitly identified maximizing as optimal from the outset.

No other effects in this analysis were significant. The proportions of dominant-color predictions across blocks for participants endorsing matching were. Mean proportions of dominant-color guesses for each of the six blocks of 50 rolls in Experiment 2 for participants in the two hint conditions who endorsed maximizing as the optimal strategy when prompted with the hint before the prediction task. In a further effort to examine the effect of a hint on performance and learning, we applied a reinforcement-learning model to all of the data from the hint feedback and no-hint feedback conditions.

We focused on these two conditions because, as can be seen in Fig. These models are alike, in that they pertain to a decision environment in which participants could rely on both advice and individual learning experience when choosing their decision strategy. The behavioral results from Experiment 2 suggest that the ARC-initial model should provide a better fit to the data because the hint advice appears to confer an early advantage see Fig.

Both models share the assumption that the participant enters the decision environment with an initial propensity for each of the two response options. Upon choosing a response option, the accuracy of the choice is used to update the propensity of the chosen option. Independent of the choice made, differential propensities to choose one response over the other decay with time.

Response sensitivity updating in response to feedback and decay rates are free parameters in both versions of the model. ARC-initial allows for an initial propensity favoring the optimal response option. ARC-outcome does not allow for an initial propensity favoring the optimal response option. A more detailed description of both models may be found in the Appendix. Individual parameter estimates were obtained for both the ARC-initial and ARC-outcome models for each of 52 participants 26 per condition.

Full details of the model fitting may also be found in the Appendix. Most importantly, only the ARC-initial model successfully converged i. The fact that only the ARC-initial model converged successfully lends further credibility to our claim that the effects of the hint are most pronounced at an early stage of learning.

This signifies that participants in the hint condition were more sensitive to differences in choice propensities. Participants in the hint condition, then, responded in a slightly more consistent—and hence, rational—fashion, given that the higher value indicates more maximizing. This makes sense, because feedback was provided in the same fashion in both conditions. Moreover, the parameter estimates for this model differ in sensible, interpretable ways across the conditions. It is, therefore, perhaps unsurprising that while the differences in parameter values were consistent with our account e.

In Experiment 1 , we found that the provision of a hint about the earning potentials of two different strategies had similar facilitative effects on maximizing rates cf. The results of Experiment 2 , however, suggested a somewhat more complicated interplay of feedback and strategy availability: Although the hint served to make maximizing more available in early trials, as the opportunities for learning increased i.

This influence extended to those participants who had correctly identified maximizing as optimal when prompted with the initial hint see Fig. The results clarify the roles of two sources of information that can help guide participants toward optimal responding in the die task. When a hint is provided, it pushes some of these participants toward maximizing from the outset see Figs. Importantly, this period of discovery takes time; the effect of outcome feedback was not apparent within the first 50 trials of either experiment.

The application of a reinforcement learning model to our data supports these general conclusions. The better-fitting model was one that combined an initial propensity to choose the maximizing response outcome, which was stronger in the hint than in the no-hint condition, with a mechanism that learned gradually from feedback. Another consistent pattern from these experiments is that when neither a hint nor feedback is provided, in the aggregate, many participants are prone to start with, and stay with, the matching strategy.

One possible interpretation of this result suggested by a reviewer is that participants, in this condition in particular, misapprehend the requirements of the task see, e. Specifically, they might believe that they are being asked to produce a potential outcome distribution for the die rolls, rather than to make independent predictions for each trial. If this were the case, probability matching would be an appropriate strategy. While this is possible, we think that this interpretation is unlikely for several reasons.

First, the incentive structure of the task payment for each correct prediction clearly emphasizes the importance of independent predictions—and participants were reminded of this structure after every ten trials in Experiment 2. Thus, the observed irrationality cannot purely reside in a failure to understand the task requirements. Over the years, many other reasons have been proposed for the persistence of probability matching despite its suboptimality, some of which may be applicable to our data.

Perhaps one of the most enduring is that the utility of predicting the rare event might outweigh the often small monetary benefit of maximizing, and might alleviate the boredom of continually making the same response e. Naturally, though, even without the prompts provided by the hint and feedback, some participants do identify maximizing as the superior strategy. The presence of such individuals suggests that some fundamental cognitive abilities over and above the impact of the independent variables contribute to maximizing behavior in binary prediction.

Taken together, these findings suggest that, for some individuals, the fully described die problem indeed yields maximizing, as would be expected from a rational agent. The ARC-initial and ARC-outcome models share the assumption that the decision maker DM enters the decision environment with an initial propensity for each of the two response options, q 1 t and q 2 t.

Upon choosing a response option, the accuracy and, as such, the reward of the choice is used to update the propensity of the chosen option. Independent of choice, q 1 t and q 2 t decay with time. The probability to choose either of the two response options is defined as the odds of a function of q 1 t and q 2 t see Eq.

After making a response, q 1 t and q 2 t are updated according to. The quantity r t is the reward received if the option was chosen, and zero if the option was not chosen. The choice propensities determine the probability of choosing the optimal response i.

For each parameter, we ran three separate chains, each of which consisted of 2, iterations, of which the first 1, were treated as burn-in samples. We examined whether the models converged successfully i. R -hat is a statistic that compares the sample variances between separate chains to the sample variances within the chains.

When the chains are indistinguishable, so are the between- and within-sample variances, and R -hat equals 1. A guiding principle is that an R -hat higher than 1. For the ARC-initial model, seven of the parameters had an R -hat higher than 1.

In contrast, 80 out of the parameters of the ARC-outcome model had an R -hat higher than 1. Footnote 6. The next step in model evaluation was to assess model fit. As a means to assess model fit, we generated model predictives of the data based on the posterior distribution and compared these to the actual data. Using these so-called posterior predictives , we calculated the proportions of times the dominant color was predicted to be chosen for each participant.

Doing so led to mean proportional discrepancies in dominant-color responses of 2. On the basis of the combination of model convergence and model fit, our preferred model of choice was the ARC-initial model, which both converged successfully and fit the data very well, as compared to the simple heuristics. It is important to note that in the tasks that we considered, the response options were negatively correlated : On any given trial, if one option delivered a reward, the other did not.

This contrasts with the preparation used in many nonhuman animal-learning studies, in which the options have been independent i.

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The Vikings would be 2. Fractional Odds are used primarily in the UK and Ireland. Few bettors use fractional odds for betting sports other than horse racing , because the conversions to understand return are difficult. To calculate winnings on fractional odds, multiply your bet by the top number numerator , then divide the result by the bottom denominator. Odds correlate to the probability of a team winning, which is the implied probability.

A favorite has about a To calculate implied probability, use the following formulas:. As a responsible bettor, it is important to understand proper bankroll management. Your payout includes your potential winnings, plus whatever you bet originally.

Sports Betting. Best Books. Otherwise put, it is four times more likely that it will rain than stay sunny. Because circumstances may change spontaneously, odds may change as well. They are not an exact science. Read sporting odds as the likelihood that one team, athlete, or horse, will win. The most common use of odds is found when placing a bet on a sporting event.

Betting agencies use historical data and team statistics to predict who is more likely to win. Whoever has the highest odds is considered the "favorite. Remember that lower odds return a higher profit. Betting on the underdog is riskier than betting on a favorite, but a higher risk means a higher potential reward. The "longer the odds," or the less likely, the more money you could win.

Learn the vocabulary of odds when betting. Many racetracks and betting establishments will have a booklet or pamphlet helping you learn terminology, but you should understand the lingo before you read odds. Some of the basic words include: Action : A bet or wager of any kind or amount.

Bookie : Someone who accepts bets and sets odds. Chalk : The favorite. Hedging : Placing bets on the team with the high odds, and the low odds, to minimize loss. Line : On any event, the current odds or point spreads on the game. Wager : The money you pay, or risk, on an outcome or event. Part 2 of Know that odds at the track tell you amount of profit you will make per dollar spent. To determine profit, multiply the amount you bet by the fraction.

Understand that fractions greater than one mean a team is an underdog. This makes sense, because you would expect a bet on the underdog to have a higher payout. If you have a hard time with fractions, then see if there is a larger number on top then on bottom. When you bet for the underdog, it is called betting "against the odds. Part 3 of Know that moneyline bets only concern what team will win the game. Odds are presented as a positive or negative number next to the team's name. A negative number means the team is favored to win, while a positive number indicates that they are the underdog.

This means the Cowboys are the favorites, but pay out less money if a bet on them wins. Try out an online to check your math when you first get started. Soon enough it will be second nature, but for now ask a friend or search for a calculator that fits your betting needs. You also get the money you bet back. To calculate how much profit you make per dollar spent, divide the amount you are going to spend by Multiply this number by the moneyline to see your potential profit.

When betting on the favorite, you take less risk, and thus earn less. Like positive odds, you earn back your bet when winning. To calculate profit, divide by the moneyline to find out the profit made per dollar spent. Part 4 of Notice that point spreads adjust the score for the favorite team. This is easiest to see with an example: If the New York Knicks are playing the Boston Celtics, and Boston is favored to win by a 4-point spread, then a bet on Boston only pays out if Boston wins by more than 4 points.

A bet on New York pays out if New York wins or if they lose by less than 4 points. If the favorite wins by the spread exactly, it is called a "push" and all bets are refunded. In the example, if Boston wins , then it is a push and no one collects a profit. If you see "half-odds" a 4. When the spread is small, moneyline bets are often better since the spread does not indicate a clear underdog. Ask your bookie about the "vig," which determines your potential profit.

Also known as the "juice," the vigorish is the commission charged for placing a bet. Typically the vig is , and you read this number like a moneyline bet see above. Sometimes there are different vigs for each team. Part 5 of If the score is exactly what the bookies set, then the bet is a push and everyone gets their money back. Make sure to check this with your bookie first, however. The "" means that a football team is favored to win by 13 points. For you to win the bet, the team must win by more than 13 points.

Not Helpful 5 Helpful Not Helpful 11 Helpful The should read Not Helpful 10 Helpful Not Helpful 23 Helpful Not Helpful 13 Helpful Not Helpful 1 Helpful 6. When I see a whole number alone on an odds sheet, what does it mean? Multiplying your stake by decimal odds gives your total return, not your profit which is total return -stake. To get to fractional from decimal, add 1. Not Helpful 38 Helpful Not Helpful 11 Helpful 7.

Not Helpful 46 Helpful 8. Not Helpful 79 Helpful 6. Unanswered Questions. Include your email address to get a message when this question is answered. By using this service, some information may be shared with YouTube. Betting through bookmakers is illegal in the United States except in the state of Nevada. It is legal in Great Britain and other countries, where it is regulated. In some countries, bookmaking is only performed by the government. Bookmakers may also take bets on non-sporting events, such as political elections.

Helpful 31 Not Helpful The money line is a simple wager in which the point spread is not determined. It is based on the odds each side has to winning.

SPORTSPUNTER SPORTS BETTING BASEBALL SOUTH KOREA

There could be a chance Skipper sees game day activation for the first time since The Houston Texans continue to make changes to their coaching staff under new coach David Culley. The year-old will be under contract with the Texans for the next three years.

Campen doesn't have an overt connection to Culley, but he was on the staff with the Chargers last year when current quarterbacks coach Pep Hamilton served in the same capacity for Los Angeles. There are some things the Houston Texans didn't lose in their trade with the Miami Dolphins to acquire left tackle Laremy Tunsil. The waiver wire system works much like the NFL draft order, except the placement can't be traded away.

As a result, the Texans will have the third crack at any player that is placed on waivers. The Houston Texans will have decisions to make regarding roster construction with new general manager Nick Caserio and new coach David Culley. Houston has 21 free agents set to hit the open market. The deals won't become final until the start of the new league year in March. The first is an unrestricted free agent. Those are players who have four accrued seasons, and are free to sign with You'll now receive the top Texans Wire stories each day directly in your inbox.

Please enter an email address. Something went wrong. October 15, The former offensive lineman for the New Orleans Saints and Green Most probability-matching experiments have used paradigms in which participants had to learn, over successive trials, the contingencies associated with each option e. Footnote 1 However, some recent studies have investigated probability matching in tasks in which the outcomes and their probabilities of occurrence are fully described to participants e.

The finding that probability matching is common even in these situations is remarkable, given that the described problems provide all of the information necessary for rational responding i. Consider, for example, predicting the outcome of rolls of a ten-sided die with seven green sides and three red sides, with a fixed payment for each correct prediction. Here there is no ambiguity about the optimal strategy always predict green , no need for a period of experimentation or exploration of the environment to determine which option is best, and no need to consider the possibility that outcome probabilities will change across successive trials.

Thus, from a normative perspective i. Newell and Rakow examined a problem like the one described above, in which participants predicted the outcomes of rolls of a simulated die. Their experiment confirmed that some participants adopted a matching strategy from the outset.

However, the experiment also highlighted a further intriguing finding: Those participants who saw the outcome of each roll after making a prediction feedback condition demonstrated an increasing rate of maximizing across the trials.

By contrast, those who received no feedback continued to match, or slightly overmatch, the objective probabilities throughout the entire sequence of trials. The facilitative effect of feedback was observed despite the feedback being completely uninformative, and therefore normatively irrelevant—the die was fair, the probabilities were stationary, and the relevant outcome probabilities were already precisely known before any feedback was received.

The finding is only puzzling, however, if one presupposes that the optimal maximizing strategy is readily generated by the decision maker in response to the initial description of the choice problem. At least for some people, the maximizing strategy might not come immediately to mind.

Specifically, before making their choices, the matching and maximizing strategies were described and participants were asked which strategy was likely to earn more money. Those given the hint question subsequently made significantly more maximizing choices than those who had not been. In short, even in a described problem in which participants have all of the information needed to identify the maximizing strategy as superior, that strategy may simply fail to come to mind. The observed impact of the hint manipulation implies that probability matching comes to mind more readily than maximizing as a candidate choice strategy in response to the problem description.

That is, outcome feedback may encourage monitoring of the reward rate for the currently used strategy as well as a search for alternative strategies that might increase payoffs. Indeed, a number of researchers have recently attempted to characterize the processes by which participants use outcome feedback to search for an optimal strategy in the binary prediction task e.

If the impact of feedback was to spur the generation of alternative strategies, which eventually led to identification of maximizing as the optimal response, then we should see an interaction whereby little or no effect of feedback would be found among those participants given the hint for whom the maximizing strategy was already highly available. This should be especially true of the subset of participants who, when prompted with that hint, correctly identified the maximizing strategy as superior.

An alternative possibility was that the hint effect observed by Koehler and James would not generalize to tasks involving trial-by-trial predictions with feedback. In their studies, the participants only made predictions for ten rolls, and did so without any information about the outcomes.

In a feedback version, it is possible that seeing the regular occurrence of the low-probability outcome might make it difficult to stick to the maximizing strategy, even if, when prompted, one could correctly identify it as optimal cf. Goodnow, The contribution of our new experiments was that they allowed us to examine the joint influences of two manipulations—feedback and provision of a hint—that up to now have only been studied in isolation.

The principal differences between Experiments 1 and 2 were the use of a real Exp. The real die was used in Experiment 1 to counter the criticism that participants might make suboptimal predictions in virtual tasks because they doubt that the die is fair and that the probabilities are stationary e.

In Experiment 1 , participants saw the same physical die being rolled on every trial, thus presumably eliminating, or at least substantially reducing, any skepticism about the task parameters. A total of undergraduate students from the University of Waterloo participated in return for course credit and earnings from the experiment.

They were randomly assigned to one of the four experimental conditions with n per condition varying from 29 to The data from eight additional participants were excluded from the analysis: five who had completed a similar task before, two who failed to follow the instructions, and one who was red—green colorblind. Participants predicted the outcomes for a series of 50 rolls of a ten-sided die with seven green sides and three red sides.

The experimenter, who was present when the predictions were made, rolled the die to determine the outcomes. This was done either after each prediction feedback condition or after all 50 predictions had been made no feedback condition. The outcomes were recorded directly on the prediction sheet, and the total winnings were calculated and paid at the end of the session.

Before completing the prediction task, one group of participants hint condition read a description of two possible strategies that could be used in the game. Which strategy do you think will win more money? Footnote 2. Error bars indicate SEM s. There was a slight tendency for predicting the dominant color more often in the feedback than in the no-feedback condition M of 8. The proportions choosing maximizing were virtually identical in the hint no-feedback The upper panels of Fig. The figures show that not all participants followed the strategy that they had just endorsed when making their trial-by-trial predictions.

However, a comparison of the upper two hint and lower two no hint panels shows that a much higher percentage of participants endorsed maximizing in the hint condition black bars, upper panels than spontaneously chose green for all ten rolls of the first game in the no-hint condition far right open bars, lower panels. This pattern is consistent with the idea that the hint prompts participants to realize that maximizing is optimal.

The mean number of green predictions across all trials for participants who identified maximizing matching as superior was 8. The key finding of Experiment 1 is that providing a hint designed to make the maximizing and matching strategies equally available increased levels of maximizing on the subsequent prediction task. In contrast, provision of feedback about the outcome of each die roll did not significantly increase maximizing.

One difference, however, between Experiment 1 and the previous research was the relatively small number of prediction trials. Newell and Rakow found that significant increases in the rates of maximization only occurred after or more prediction trials. This allowed us not only to attempt to replicate the findings of Experiment 1 and of Newell and Rakow, but also to examine the interplay of the hint and feedback over an extended number of predictions.

The increased number of training trials also facilitated the application of reinforcement learning models to our data. These were applied in an effort to find converging evidence of the roles played by the hint and feedback in the binary prediction task. In the two feedback-present conditions, an image of a ten-sided die rolled across the screen following each prediction and the outcome was displayed.

In the two feedback-absent conditions, no die was shown and the participants were told that the die would be rolled once all of the predictions had been made, in order to determine payment. The dominant color i. Mean proportions of dominant-color guesses for each of the six blocks of 50 rolls in Experiment 2. To provide a like-for-like comparison between Experiments 1 and 2 , we analyzed the initial 50 trials of Experiment 2 i.

This replicates the results of Experiment 1 for the equivalent number of trials. It shows that the change from a real to a virtual die, the change in population Canadian vs. Australian participants , and the absence of the experimenter in Experiment 2 did not significantly alter the results. Neither of these individual block analyses showed significant interactions between hint and feedback. In short, the hint manipulation had an influence on early prediction trials but not on later ones, leading to no overall main effect of hint.

This effect is especially prominent in the no-hint no-feedback group, in which matching was overwhelmingly dominant in the first ten-roll game lower right panel. Numbers of dominant-color guesses made by participants in the first ten-roll game of Experiment 2. To investigate in more detail the interplay of feedback and hint over the entire trials, we examined the prediction data across blocks for the subset of participants in the hint conditions who explicitly identified maximizing as optimal from the outset.

No other effects in this analysis were significant. The proportions of dominant-color predictions across blocks for participants endorsing matching were. Mean proportions of dominant-color guesses for each of the six blocks of 50 rolls in Experiment 2 for participants in the two hint conditions who endorsed maximizing as the optimal strategy when prompted with the hint before the prediction task. In a further effort to examine the effect of a hint on performance and learning, we applied a reinforcement-learning model to all of the data from the hint feedback and no-hint feedback conditions.

We focused on these two conditions because, as can be seen in Fig. These models are alike, in that they pertain to a decision environment in which participants could rely on both advice and individual learning experience when choosing their decision strategy. The behavioral results from Experiment 2 suggest that the ARC-initial model should provide a better fit to the data because the hint advice appears to confer an early advantage see Fig.

Both models share the assumption that the participant enters the decision environment with an initial propensity for each of the two response options. Upon choosing a response option, the accuracy of the choice is used to update the propensity of the chosen option. Independent of the choice made, differential propensities to choose one response over the other decay with time.

Response sensitivity updating in response to feedback and decay rates are free parameters in both versions of the model. ARC-initial allows for an initial propensity favoring the optimal response option. ARC-outcome does not allow for an initial propensity favoring the optimal response option. A more detailed description of both models may be found in the Appendix. Individual parameter estimates were obtained for both the ARC-initial and ARC-outcome models for each of 52 participants 26 per condition.

Full details of the model fitting may also be found in the Appendix. Most importantly, only the ARC-initial model successfully converged i. The fact that only the ARC-initial model converged successfully lends further credibility to our claim that the effects of the hint are most pronounced at an early stage of learning.

This signifies that participants in the hint condition were more sensitive to differences in choice propensities. Participants in the hint condition, then, responded in a slightly more consistent—and hence, rational—fashion, given that the higher value indicates more maximizing.

This makes sense, because feedback was provided in the same fashion in both conditions. Moreover, the parameter estimates for this model differ in sensible, interpretable ways across the conditions. It is, therefore, perhaps unsurprising that while the differences in parameter values were consistent with our account e. In Experiment 1 , we found that the provision of a hint about the earning potentials of two different strategies had similar facilitative effects on maximizing rates cf. The results of Experiment 2 , however, suggested a somewhat more complicated interplay of feedback and strategy availability: Although the hint served to make maximizing more available in early trials, as the opportunities for learning increased i.

This influence extended to those participants who had correctly identified maximizing as optimal when prompted with the initial hint see Fig. The results clarify the roles of two sources of information that can help guide participants toward optimal responding in the die task. When a hint is provided, it pushes some of these participants toward maximizing from the outset see Figs. Importantly, this period of discovery takes time; the effect of outcome feedback was not apparent within the first 50 trials of either experiment.

The application of a reinforcement learning model to our data supports these general conclusions. The better-fitting model was one that combined an initial propensity to choose the maximizing response outcome, which was stronger in the hint than in the no-hint condition, with a mechanism that learned gradually from feedback.

Another consistent pattern from these experiments is that when neither a hint nor feedback is provided, in the aggregate, many participants are prone to start with, and stay with, the matching strategy. One possible interpretation of this result suggested by a reviewer is that participants, in this condition in particular, misapprehend the requirements of the task see, e.

Specifically, they might believe that they are being asked to produce a potential outcome distribution for the die rolls, rather than to make independent predictions for each trial. If this were the case, probability matching would be an appropriate strategy. While this is possible, we think that this interpretation is unlikely for several reasons. First, the incentive structure of the task payment for each correct prediction clearly emphasizes the importance of independent predictions—and participants were reminded of this structure after every ten trials in Experiment 2.

Thus, the observed irrationality cannot purely reside in a failure to understand the task requirements. Over the years, many other reasons have been proposed for the persistence of probability matching despite its suboptimality, some of which may be applicable to our data. Perhaps one of the most enduring is that the utility of predicting the rare event might outweigh the often small monetary benefit of maximizing, and might alleviate the boredom of continually making the same response e.

Naturally, though, even without the prompts provided by the hint and feedback, some participants do identify maximizing as the superior strategy. The presence of such individuals suggests that some fundamental cognitive abilities over and above the impact of the independent variables contribute to maximizing behavior in binary prediction.

Taken together, these findings suggest that, for some individuals, the fully described die problem indeed yields maximizing, as would be expected from a rational agent. The ARC-initial and ARC-outcome models share the assumption that the decision maker DM enters the decision environment with an initial propensity for each of the two response options, q 1 t and q 2 t. Upon choosing a response option, the accuracy and, as such, the reward of the choice is used to update the propensity of the chosen option.

Independent of choice, q 1 t and q 2 t decay with time. The probability to choose either of the two response options is defined as the odds of a function of q 1 t and q 2 t see Eq.

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