CS 286r Comments
10/15/2008

Malvika Rao	Non-myopic strategies in prediction markets: This paper examines a simple model with two traders in a prediction market setting. It discovers that myopically optimal strategies do not form an equilibrium under the logarithmic market scoring rule. The paper makes the assumption that signals are conditionally independent. However bluffing can be reduced by introducing a discounted market scoring rule which makes future non-myopic payoffs (obtained through bluffing) less valuable. Their results appear to generalize to multiple traders and signals. It is unclear whether bluffing would be more prevalent under this model in a real world setting. If player 1 can get a higher payoff through bluffing then player 2 can guess this. Hence by the common knowledge assumption player 1 may know that player 2 knows etc. This involves a considerable amount of strategizing and it is not obvious that the payoffs obtained through bluffing are sufficient to warrant the effort to strategize. An empirical measure that sheds light on the likelihood of bluffing vis-a-vis the payoff scheme would be useful. A behavioral analysis of what is the equilibrium that is most likely to be played in reality would be interesting to know. It would also be interesting to determine when myopic strategies would form a better equilibrium despite the presence of non-myopic strategies.
Rory Kulz	Non-myopic Strategies in Prediction Markets After I read the papers last time and submitted my comments, I hadn't yet read the Pennock blog entry, wherein I discovered (to my admitted partial surprise) that the logarithmic market scoring rule (LMSR) was indeed in use out in the wild. Despite being in operation in the real world, an important unanswered question is: is market manipulation possible and to what extent? This paper tackles that for prediction markets based on the LMSR. It was already known that for agents ignoring the impact of their reporting on potential future profits (i.e. myopic agents), reporting honestly is an optimal strategy. It turns out that not only might non-myopic agents be able to game the market by misleading other traders then profiting by correcting the market but that the optimal myopic strategy is not a weak PBE strategy for the LMSR. However, with some slight modification, the incentives for this behavior can be attenuated. This paper uses more reasonable assumptions I think than the papers they detail in the related work section, but the discounting technique presents some issues that are not really expanded on. While it's interesting, has *this* been implemented out there in the real world? They acknowledge that a high discount rate may "dissuade traders from participating in the market;" isn't this particularly problematic for predictions which take place over a long period of time? The possibility of characterizing when myopic strategies can be in equilibrium seems exciting, yes, but probably only tractable in some specific cases. The properties of the discounting scheme beg the question of other possible modifications to the LMSR which might alleviate the problems of market manipulation. Bluffing and Strategic Reticence in Prediction Markets I read this paper after the Dimitrov-Sami paper. Like the first paper, it examines the problem of players not playing myopic truthful strategies in prediction markets. I'm having trouble seeing exactly where the argument breaks so that this paper and the previous arrive at totally different conclusions regarding the logarithmic market scoring rule mechanism. I could really benefit from a sort of walkthrough of the two results and why conditional independence is such a stronger condition to place on the agents. It was interesting to read about the dynamic parimutuel market mechanism and see how it functions. It seems a little more natural, a little more intuitive, and less a product of academics than Hanson's. Are there other, less well-known mechanisms that have been explored for aggregating information?
Ziyad Aljarboua	Bluffing & Strategic Reticence in Prediction Markets: This paper examines behavior of traders and their tendency for bluffing and reticence. specifically, this paper examines conditionally independent signals and conditionally dependent signals structures for logarithmic market scoring rule with risk neutral traders with incomplete information. It is shown in this paper that when signals of traders are independent conditional on the state of the world, traders' betting is a perfect Bayesian equilibrium and when signals are conditionally dependent, there is a joint probability distribution on signals. In the conditionally independent signals, a 3 stage game with 2 risk neutral players each has a private signal is considered. The LMSR market is aimed at prediction the true state of W. The proof in the paper shows that the first player truthfully reports their probability in the first stage of the game. The second player also reports their true probabilities. Finally, the first player plays again and makes no change since all information have been reveled by the end of the second stage of the game. So, its clear from this example, that all agents act truthfully. This 3-stage, 2 player game can be extended to finite player finite stage game as shown in the paper. In the Conditionally dependent signals, an example where two players participate in a LMSR prediction market based on a coin flip is considered. It is pr oven in the papers that both players have the incentive to bluff. It is shown that players can affect probability by bluffing. When traders anticipate sufficiently better-informed traders entering the market in the future, they are incentivized to partially hold their information by either not participating in the market or by moving the market probability toward their beliefs. This suggestion is enforced by the no trade thermos which state that rational, risk neutral agents will not trade at all when interacting in a zero sum market. This is because trading in the market will signal to other agents and discourage him/her from trading. In conclusion, it is clear that conditionally independent signals has a perfect Bayesian equilibrium where players truthfully reveal their private information at their first Chance. And with conditionally dependent information, there exist a joint probability distribution where traders have incentive to bluff or bet against their own information to mislead other traders for a long term goal of correcting the price. Non-myopic Strategies in Prediction Markets: This paper analyzes non-myopic strategies and myopic equilibria. Trading strategies in the LMSR prediction market are studied to model a general Bayesian framework. Also, a simple discounted market scoring rule is proposed to reduce the opportunity for bluffing strategies. the model considered in this paper consists of 2 risk neutral players with private information. This model is specified by only four probabilities. in this model, traders make market moves and report probability at true end. It is shown that if players are restricted to myopic behavior, all information will be aggregated in just two trades and the market will perform well. this is true since it is optimal for a trader o report their true belief about the likelihood of an event as log as the impact of their report on the their profit from future trades is ignored. In the case where players are not restricted to the myopic behavior, where players my deviate from their strategies to exploit other player's actions, players are incentivized to bluff depending on the profitability of the new strategy compared to the honest strategy. This paper shows that the bluffing strategy is always more profitable than the myopic strategy under the informativeness condition. Thus, traders are incentivized to bluff. the second part of the paper is a rule, the discounted market scoring rule, where traders' payoffs are discounted over time to reduce the potential gain from correcting a misled trader. The motivation for this model is the fact that traders exploit the market by misleading other traders about the true value of a security to benefit in later trades leads traders to not make any predictions based on the market price. In conclusion, it was shown in this paper that the myopically optimal strategy profile does not constitute a weak perfect Bayesian equilibrium strategy for the LMSR even for multiple traders and signals. Also, myopic strategy are normally not in equilibrium when non-myopic strategies are admitted. There are some limitation to this conclusion. Traders might prefer the honest optimal strategy in real market for other reason. Also, the balance between the payoff by bluffing and simplicity of the bluffing strategies are not considered in the paper. This fact might make the need for such proposed model nonexistent.
Hao-Yuh Su	1. Non-myopic Strategies in Prediction Markets 2. Bluffing and Strategic Reticence in Prediction Markets. Each of these two papers discusses about the opportunity of bluffing strategies in Logarithmic Market Scoring Rule (LMSR). Based on different information structure, the two papers have completely opposite results. The first one is under the condition of independently generated signals, and concludes that there is no finite equilibrium (no weak PBE) in the prediction market. It is shown that myopically optimal strategy profile is not a weak Perfect Bayesian Equilibrium (PBE) strategy for LMSR. Moreover, it came up with an idea of discounted market scoring rule, which eliminates the bluffing strategy and makes the market price converge to optimal price. In the 2nd paper, it is shown that with conditionally independent signals, players are better off to play truthfully through the betting process; however, with conditionally dependent signals, it's better for a player to hide her information, misleading others and make a profit by correcting the errors in later rounds. Both of these two papers are highly important. It is essential to examine the existence of finite equilibrium of the market scoring rule, and it is also important to inspect the possibility of manipulation under any given condition. I'm quite surprised that giving different information structures will lead to two opposite results. Before I make any comment on it, I want to make several things clear first. The first question is that what exactly is so-called "conditionally independent?" Can I have a definition and a clear example for it? The second question is that is there any other information structure existing? If so, I believe it's a good topic to reexamine LMSR under different situations. I think some of the ideas in these two papers are similar to the 3rd question (3b) in Assignment 3. Because the 2nd player will acquire the whole information and move to the optimal price if the 1st player plays truthfully, there is always an incentive for the 1st player to bluff. Therefore, "the firm" in (3b) will never stick to the corresponding campaign budget directly derived from the cost of the product in order to prevent the consumers from calculating the reasonable price and then loss a chance to make a profit from overpricing. As for the applications of these two papers, beside to reexamine LMSR under different premises, we can also apply the same technique to inspect other market scoring rules. For example, we can examine the dynamic pari-mutuel markets (DPM) with the procedure in the 1st paper.
Brian Young	Non-Myopic Strategies in Prediction Markets (Dimitrov, Sami) Bluffing and Strategic Reticence in Prediction Markets (Chen, Reeves, Pennock, Hanson, Fortnow, Gonen) comments by Brian Young Both Wednesday's papers deal with the issue of strategic playing in prediction markets that use market scoring rules. The "myopic" strategy, in which a participant bases her honest choice on her true beliefs, may be suboptimal compared to the "bluffing" strategy, in which the participant attempts to convince other players that her beliefs are different from what they really are. The Dimitrov and Sami paper concludes that an optimal price is reached by a discounted market scoring rule, where over time, the value of a trade goes down, such that later trades are discounted more than earlier ones. Again, it is not intuitive to me how such a rule might be translated into a prediction market. In addition, I wonder how this result could be adapted to a market situation in which players do not have to strictly alternate moves, particularly with the addition of other players. In addition, the papers find that the ways in which markets are modeled lead to substantially different results. How might we refine our ideas of which models are most appropriate, for instance, whether signals can be modeled as conditionally independent or conditionally dependent?
Avner May	Non-myopic Strategies in Prediction Markets (Stanko Dimitrov, Rahul Sami) Bluffing and Strategic Reticence in Prediction Markets (Chen, Reeves, Pennock, Hanson, Fortnow, Gonen) These two papers tackled an interesting question: are there situations in which traders in prediction markets have incentive to withhold private information, or intentionally mislead other traders in order to eventually profit in the end? This question is particularly important given the hope that these prediction markets should theoretically be a perfect aggregation of everyone’s perfect information. I thought that the Dimitrov-Sami paper’s way of framing the problem was very effective: there are two players (or in general, m players), each gets a binary signal, and the best probability estimates are a function of the two signals received by the players. I found this to be similar (though a more general formulation) of the coin flip scenario (although in the coin flip scenario the signals are precisely what is being bet on, which is a significant difference). It provides good intuition for why people could be motivated to bluff at the beginning stages of these “games”. I thought that introducing a discounting factor by Dimitrov-Sami was a clever way of providing incentive to traders to reveal their information early, as opposed to bluff. I was confused by the difference between the different types of signals – eg, independent vs. conditionally independent, and why this was so central to the different results in these two papers; what exactly is the correlation between a signal and the eventual outcome? One question I have is how the belief that Player 1 may or may not bluff would affect the actions of Player 2; in these papers, in order to test whether strategy profiles were equilibriums, the authors generally assumed that Player 2 would act myopically in response to Player 1. In general, I think that these papers together provide a good survey of the potential downfalls to these prediction markets through bluffing and information withholding. One downfall to this topic seems to be that it is very theoretical; these papers do not provide empirical data to support their claims, and the authors seemed to express that testing any of these theories in real market situations would be difficult. Are these untestable theories? If so, what exactly is the value they provide?
Zhenming Liu	Both papers exam whether being dishonest will benefit the participants in prediction market. The fact that subtle differences in the model leads to opposite result is actually quite astonishing, as [DS] declared. I believe the reason of building mathematical models in social science is trying to capture the features of a system to enable rigorous analysis while ignoring the irrelevant. Clearly, both [CRP] and[DS] provide very neat model on prediction market. And the fact that these two models result in opposite conclusion suggests our view of prediction market is quite incapable of “tolerating noises”, in the sense of that a small twist of assumption set dramatically changes the behavior of the market. Therefore, I strongly agree [DS]’s claim in their conclusion section that a full characterization of the information structure on which myopic strategies are in equilibrium are exciting. I particularly like [DS]’s approach of introducing the notion of discounted scoring rule. This reflects two important spirits of computer science and engineering---modularization and “localized debugging”. The name “discounted scoring rule” already suggests that this approach is a combination of two modules: “discounted utility function” and “log scoring rule”. Readers can easily understand this combination probably have advantages from “discounted utility” and “log scoring rule”. Realizing the non-myopic behavior, [DS] found a “hot fix” to the mechanics in the prediction market. This is the most satisfying way of “debugging” a system. [DS]. S.Dimitrov, R.Sami, “Non-myopic Strategies in Prediction Markets” [CRP].Y.Chen, D.M.Reeves, etc “Bluffing and Strategic Reticence in Prediction Markets”
Victor Chan	Victor Chan Paper Comments: Non-myopic Strategies in Prediction Markets & Bluffing and Strategic Reticence in Prediction Markets The main contribution of the two papers by Dimitrov & Sami, and Chen et al. dealt with the role of non-myopic strategies in prediction markets. The two papers are closely related in evaluating whether bluffing or holding information are part of the perfect Bayesian equilibrium strategies. This study is important, because it gives insight to whether or not traders will have reach a higher payoff it they purposely mislead others in the market during the game, by not releasing their on information. Furthermore this is important because, prediction markets would be more accurate, if every player played a myopic strategy, rather than trying to manipulate the market for higher profits. The result of the two papers were actually quite different, and this was the result of the use of different information structures. In Dimitrov's paper, the information structure for a player was basically a independent bit and randomly choosen. Using this structure and evaluating the game, it is shown that myopically optimal strategy is not an equilibrium. In contrast Chen et al. found that a myopic strategy was in fact the equilibrium strategy, and the difference arises as a result of using an information structure which is conditionally dependent on the global state. Finally Dimitrov's paper deals with a infinite step game, while Chen's paper deals with finite stage games. Dimitrov's paper also looked into discounting the payoff of trades as time passes. This was studied to help decrease the incentive for traders to mislead others, then trying to gain a profit in correcting the mistakes in the market. While, Chen et al. also found that if the information signals were dependent it would result in non-myopic strategies being the equilibrium strategy. Some of the things that were unclear include why the difference in the information structures lead to such different results, or perhaps because of the difference in models, the two papers were evaluating different games all together? The results can be extended to show the effects of non-myopic strategies on the overall accuracy of the prediction market. ie How does bluffing effect the accuracy over time. I think an interesting project would be to monitor a prediction market and have traders report anonymously on whether or not they are bluffing, and whether or not they think others are bluffing. An empirical study might yield more interesting results and show how a player information signal affects their decision in choosing myopic or non-myopic strategies
Nikhil Srivastava	Dimitrov and Sami prove that under LMSR and certain further criteria, myopically optimal strategies (the only ones considered in our earlier papers) in fact are not weak PBE strategies. This is identified in the paper by considering the possible advantages of "bluffing" - executing dishonest trades and profiting from the resulting error in market price. A particularly important consequence is that a market conducted under these criteria is not sure to converge to its optimal price after a finite number of rounds. Finally, the authors show that discounting future payoffs causes the relative entropy between the market price and the optimal price to reduce exponentially over time. This paper examines the very important and real issue of market manipulation from a theoretical point of view, and arrives at a formal proof of the inadequacy of LSMR to prevent this undesired phenomena under certain conditions. It would be interesting to present more methods (other than discounting) that successfully bound the market error, but I imagine they are hard to arrive at or even guess. Chen et al also examine non-myopic strategies, though with different criteria and obtain very different results. Under conditionally independent signals (where signals are drawn conditioned on the state of the world), the strategy of all players trading honestly and immediately (and each believing every other player does the same) is a PBE under LSMR. With conditionally dependent signals, however, the result is the opposite - there can exist certain probability distributions under which agents have incentives to trade dishonestly. I would have expected that the incentive to bluff is lower for more players in the market? Perhaps this is not the case for zero trading cost, where infinite infinitesimal trades are valid.
Alice Gao	Both papers try to outline scenarios in which players have incentives to bluff in a market using market scoring rules. The fundamental reason for the possibility of long-range manipulation is that the MSR mechanism is not incentive compatible in general. Both papers introduced the possibilities of manipulative strategies using MSR under different assumptions. It is intriguing, as stated by Dimitrov and Sami that different assumptions can lead to completely opposite results. On one hand, I am not entirely sure if I understand the assumptions used by these two papers correctly and how they correspond to each other. On the other hand, it would be interesting to ask the question that which assumption is more realistic. In particular, which assumption would be a better approximation of the assumptions of traders participating in real prediction markets? I also noticed the general informativeness condition assumption used by Dimitrov and Sami, and I am not sure how the result would differ if this condition would be relaxed. Likewise, based on these theoretical results on trader behaviours, a further research direction would be to understand behaviours of traders participating in real markets with respect to the respective theoretical results. For example, Dimitrov and Sami mentioned a good point that in practice players might prefer myopically optimal strategies even if theory says that it is not the equilibrium behaviour. Some possible reasons are that myopically optimal strategies can be simpler to play, more robust, and the potential games from bluffing are often very small. This is also related to the question of computational cost for a trader, since some traders might prefer a simpler strategy just because they do not have sufficient technical background to figure out the more profitable long-range manipulative strategy. Moreover, both papers seem to use very simple game scenarios (small number of players, small number of rounds, deterministic order of play), and it is questionable whether these results can be easily generalized to larger and more complicated scenarios. I agree with Dimitrov and Sami that an important future research direction would be to characterize class of information structures for which myopic strategies are in equilibrium. Lastly, I found the discussion of the discounted MSR to be very encouraging since it presents a simple and effective way to incentivize players to reveal their true information early in the game. The exponential convergence of the entropy of the prices is also a very nice and strong result. This also raises the question of selecting a proper value for the discount factor. Also, I wonder what would happen if we have a market in which the trader population is not static. I imagine that the discount factor would cause the market to gradually lose incentive for further traders to join the trade as time goes on. So for the discounted scenario to work effectively, it seems that we have to assume traders who do not join the market at the beginning do not have critical information that will alter the market price by a significant amount.
Sagar Mehta	The paper by Dimitrov and Sami studies non-myopic strategies in LMSR under the assumption that signals of players are unconditionally independent and the LMSR market has infinite periods. It finds that in a two player prediction market setting, bluffing can be an equilibrium strategy. The paper by Chen et. al. compliments this research by considering games with finite periods and finite players: for both conditionally independent and conditionally dependent signals. The implications of this paper are mainly that non-myopic behavior can delay price discovery in market scoring rule prediction markets. An interesting avenue of future research could be to extend the results from the simple cases presented to more complex cases with a large number of participants. The Dimitrov and Sami paper mentions that due to the strategic complexity of double-auction markets, characterizing information scenarios where manipulation is possible is difficult. Does the same hold true for market scoring rule information markets with many participants? The ability to classify situations where bluffing or strategic reticence is an equilibrium strategy could be an interesting area of research.
Angela Ying	Bluffing and Strategic Reticence in Prediction Markets This paper discussed non-myopic betting strategies for prediction markets, but was different from the other paper in that it focused on conditionally independent, finite player, finite stage games, as well as a game with conditionally dependent signals. It consequently found two different results – for the conditionally independent games the authors found that the Perfect Bayesian Equilibrium of the game is for every player to be truthful from the beginning. For conditionally dependent signalled games, the PBE of the game involved players bluffing, since otherwise it would be very easy for a player to lie. The first result is especially significant and is very different from the result obtained from the Dimitrov/Sami paper. This is probably because we restricted the games to be finite player and finite stage. As a result, we can use a process similar to backwards induction to figure out all the information from all the players, which means that they have no incentive to lie. In an infinite game, because there may not actually be an end, there is always some uncertainty as to what the players truly believe. However, it would be interesting to see what would happen if we combined the two types of games, making a finite game with the signals being completely independent (as opposed to conditionally independent), or making an infinite game with signals conditionally independent. The latter may be difficult because being conditionally independent suggests that the play somehow converges and the prediction market has accurate information from trading. Another possible extension is to look at games using a different market scoring rule, such as a quadratic MSR. Non-myopic Strategies in Prediction Markets I thought this was a very interesting paper on strategic bluffing in prediction markets. Overall, the main contribution of the paper was its studies and theorems concerning non-myopic strategies, specifically that in a normal game with imperfect information players will always want to bluff with a non-zero probability. They prove that the myopic equilibrium in a market where players play their true beliefs and contribute full information to the market is not a perfect bayesian equilibrium because more profit is to be had by first bluffing, leading the other player to think that their information is the opposite of what it is, and then playing the real strategy to make a profit. The authors also introduced a way to prevent this from happening by making the market such that trade profit decays over time, incentivizing players to play their real strategy from the very beginning. This whole concept is interesting because of the interactions between players and the information they glean from each other. I thought this paper explained the concepts pretty well - however, I am curious to see what happens in a market where every single player bluffs. Most of the paper was devoted to explaining why playing purely myopic strategies was not an equilibrium, and it assumed in its proofs that every player except the one we were focusing on was playing myopically. If most of the people in the market bluff, does information still converge at the end? This seems reasonable, especially given the section on bounded relative entropy at the end of the paper (however, it was unclear whether this market included decaying profits?). Another thing I thought was interesting was the difference in results between this paper and the Chen paper, since conceptually, conditional independence isn't that different from independence in a situation where the information should converge anyway.
Subhash Arja	The main goal of the "Non-myopic Strategies in Prediction Markets" by S. Dimitrov and R. Sami was to describe a new way to analyze non-myopic strategies and to prove that the myopically optimal strategy profile is not a weak Perfect Bayesian Equilibrium. As noted in the paper, the limit of the research is that it doesn't take into account possibilities where the trader may profit by first misleading other traders and ten correcting their errors in subsequent rounds. The model described assumes that traders learn from prior trades as well as their own signals. This study is important and applicable to prediction markets as well as possibly financial markets, since their actions depend heavily on the actions of individuals and their bluffing and misleading tendencies. The paper provides insight into instances when market scoring rules may be greatly affected by long-term manipulations by the participants. The theoretical proofs and foundation is done through initial analysis of a simple problem involving two traders, and each trader can see the other's signals. The logarithmic scoring rule is used in this instance. One project idea to apply this study is to take one stock in a financial market and analyze how imperfect or hidden information known only to a few traders affects the growth of the stock. Manipulation can be detected by analyzing the actual worth of the stock depending on the company's performance versus the hype created by those bluffing. The paper "Bluffing and Strategic Reticence in Prediction Markets" by Y. Chen, et al., seeks to undertake a similar model as the previous paper but arrives at a different end result. Unlike in the previous paper, it is found that traders can benefit by either hiding information or bluffing. The model used here uses a logarithmic scoring rule, like the previous paper, and shows that when signals are conditionally independent on the state of the world, not bluffing is a Perfect Bayesian Equilibrium. The main contributor to the different conclusions from both papers is the difference between traders' signals being conditionally dependent or indepedent. The first paper assumes traders' signals are dependently generated, while the second paper uses a conditionally independent approach. Also, the second paper uses an example of a 3-round market, while the first paper only considers analysis of a simple model with 2 players and 2 signals.
Haoqi Zhang	The main contribution of the two papers is the characterization of perfect bayesian equiibria in prediction markets using market scoring rules when the strategy space allows agents to be forward looking. Specifically, if we allow agents to make multiple reports of their probability assessments of an event, there is the possibility of an agent reporting false information (or withholding information) for the purposes of changing another player's report so as to be able to gain from a further re-report. The two papers show that when signals are conditionally independent on the outcome, truthful betting is a PBE for log market scoring rules. Furthermore, the papers show that non-truthful betting is possible when the players' signals are conditionally dependent or the probability of the event is jointly dependent on the two events. In the case of conditional independence, given that players can infer posteriors from other player's signal, they can change the market probability to take into account everyone's assessment thus far. This is important because it shows that even if agents are forward looking, with conditional independence they will report their assessment immediately in equilibrium, allowing the market to quickly incorporate information. In the case that signals are dependent, players have incentive to misreport so as to be able to capitalize on the other players' truthful betting based on an earlier false report. I am left with a couple of questions after reading the two papers: - what does it mean for an agent to make multiple reports if he only gets information once? Is this something that we actually want to allow? - what is the risk that a player entails by misreporting (if any)? - what are the costs/limitations of implementing discounted payoffs?
Andrew Berry	Bluffing and Strategic Reticence in Prediction Markets The results of the paper are important because they give us some intuition why prediction markets may have some error when the aggregate information suggests the probability of an event occurring deviates from the true probability or true belief. I think a natural extension of the work is to determine whether other market types combined with outer scoring rules lead to similar equilibrium behavior. I also wonder what would occur if we relaxed the assumption that prior distribution over states of the world is common knowledge to all players. Is this a reasonable assumption? Especially considering the general case when there are more than two market participants, I’m not sure that this would necessarily hold. It's interesting to see that simple assumptions can have drastic effects on conclusions. Comparing the two papers for Wednesday, assuming signals conditionally independent versus assuming that signals are generated independently account for the difference in the existence of a myopic equilibrium. One thing I wish both papers would have done (particularly the second paper because it was published after Chen et al) is to have explained the independence assumption and why it is valid (conditional or not). An applied example showing the merits of each assumption would go a long way in helping me to judge which paper has more convincing results. Non-Myopic Strategies in Prediction Markets I am most curious about the assumptions surrounding the results. The main caveat for the results in this paper is that they hold for all scoring rules with convex divergence functions. So while this holds for the quadratic and logarithmic scoring rules, does this property exist in the most commonly cited/used scoring rules? Additionally, given this limitation, can the results be extended in a reasonable way? I also think that in real-world settings traders would not have initial signals that are independent and a shortcoming of the work was not explaining the rationale behind this assumption. On another note, I don't quite buy the importance to section five. Proving that under the discounted market scoring rule information aggregation does occur in the long run is a nice proof, because the paper proved in section 3 that it cannot be determined that the market will meet this convergence after n rounds, this gives us very little practically in determining what the "long run" actually is. Needless to say this only applies to the new scoring rule introduced (Although I do think the discounted scoring method is realistic).
Travis May	I intend to focus this response on the limitations of the models proposed by "Bluffing and Strategic Reticence in Prediction Markets" and "Non-Myopic Strategies in Prediction Markets." Beyond the specific constraints of these models, I would like to raise the point of a different type of strategic behavior that may manipulate markets: exogenous motives. Presumably, the market-maker, who is subsidizing the game in order to acquire information, has some proposed purpose for that information once she receives it. If her action is truly affected by what happens in the prediction market, then a person who could be affected by her action may choose to participate themselves in the market. For example, in the election, some Democratic strategists suggested that Intrade conditional probabilities of victory should be consulted before choosing a candidate. Upon making the suggestion, manipulation was visible in these probabilities, as fans of each of the candidates were vying for the non-financial prize of being the favorite candidate in the general election. In fact -- such behavior was recently incentivized, as the Presidential debates have featured a market on which candidate's Intrade price would rise more following the debate. This encouraged manipulation on all sides, both of the prices for that contract and for the candidate's contract. This is much of an aside from the market manipulation and bluffing that the two papers comment on, but warrants discussion, and it fits the broad theme of prediction markets not being able to control strategic behavior. Ultimately, perhaps the best way of detecting strategic behavior would be through an algorithm of sorts, monitoring the trades you make and ensuring they don't appear strategic in either case.
Xiaolu Yu	Bluffing and Strategic Reticence in Prediction Markets Although a prediction market might be designed to subsidize the market by employ an automated market maker, the author of this article points out that under a proper scoring rule which is incentive compatible for risk-neutral agents, when traders with conditionally dependent signals can change the possibility estimate multiple times, they might have incentives to manipulate information and mislead other traders by bluffing or strategic reticence. The author also reminds that conditional dependence of signals is not a sufficient condition for bluffing in LMSR. In contrast, playing truthfully in a multiple stage game in LMSR with conditionally independent signals is equivalent to selecting oneself as the first player, and will worse off by deviating because in every stage of the game, he/she has to pay si(rold). Most of the prediction markets, however, are not incentive compatible, or with conditionally dependent signals for players. Forward-looking traders may find out "tricks" in the game by closely and continuously observing other traders' strategies, and get an incentive to strategically mislead other traders with the intent of correcting the errors made by others in a later period of the sequential game. One of the unconsidered scenarios is that prediction markets with irrational traders. Bluffing and strategic delay may have different impact on irrational traders, and would possibly result in unexpected results for the bluffers (e.g. negative payoff). However, this may not happen most of the time. Non-myopic Strategies in Prediction Markets This paper reaches one of the same results as the previous one: a player has an incentive to not fully divulge his private information revealed by his signal when he may play multiple times in the market, which, in turn, results in nonexistence of finite informative equilibria. The author thus proposes a scheme, the discounted market scoring rule, in order to reduce the chances of traders to exploit erroneous trades induced by their untruthful bets, by discounting traders' payoffs for market transactions over time. The two papers are also consistent on that myopic strategies are generically not in equilibrium when non-myopic strategies are admitted. The contrast, interestingly, is the deduction from this point. The previous paper investigate under what information structure the myopic strategies are in equilibrium and the non-myopic strategy are preferred, whereas this paper proposes the discounted log-MSR and shows that complete aggregation of information surely happens the long run and market price will converge towards the optimal value for the particular realized set of information signals in any weak-PBE profile. One unanswered question is the convergence rate of the market price. Another problem the article doesn't address clearly is that if there are multiple players, whether the time to join the game (i.e. after how many rounds one player joins) affects payoffs.