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Non myopic strategy Truth or Lie?. Scoring Rules. One important feature of market scoring rules is that they are myopic strategy proof. That means that it is optimal for a trader to report his true belief about the likelihood of an event.
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Scoring Rules • One important feature of market scoring rules is that they are myopic strategy proof. • That means that it is optimal for a trader to report his true belief about the likelihood of an event. • But ignore the impact of his report on the profit he might get from future trades. • Hence, Bluffing first and telling the truth might be better than telling the truth!
What’s in today class? • Simple model (2 partially informed traders in single information market). • Extend results to more complicated markets with multiple traders and signals. • New scoring rule which reduces the opportunity for bluffing strategies.
Introduction • Information market gather data on future events. An informed trader can use his private information to recognize inaccuracies and make profit. • Those trades influence the trading price. • Those price provide signals to others about the future event which causes them to adjust their belief about the true value of their security.
Example • Say in the 2012 US election campaign. • When market price showed probability of 50% that either Obama or Romney will win. And, I thought Romney will win. • Day before the election I checked and saw that there is 70% that Obama will win. I tell myself if the markets thinks Obama is going to win I should change my mind and join the market opinion!
Ideals… • Ideally, this will lead to a situation in which all traders reach common consensus that reflects all available information. • Hence the conclusion that prediction markets rely on traders adjusting their beliefs in response to other traders trade (the market price)
There is your drawback • Traders can mislead other traders about the value of their security and than profit from their mistake in some later trade! • This can also cause traders to be cautious about making inferences from market prices, thus damaging the data aggregation of the market.
2-players and 2 signals • Need to predict future event E. • Two players: P1, P2. each has private info about E.(say x1,x2) • Assume traders share the prior probability distribution.
More assumptions • The two signals are independent. • X1 is 0,1 with prob 0.5,0.5 • X2 is 0,1 with prob q, 1-q (0<q<1) • E(E/x1x2=00) = p00, E(E/x1x2=10) = p10 and so on. • The model can be fully specified:
Trades • Trade in security F base on prediction of event E is done using a market scoring rule. • Players make sequence of market moves. • In each move the player announces the a probability Pi for the event E. • In the paper they look on logarithmic scoring rule.
Let’s play • Suppose P1 saw x1 = 1. and he is myopic. She will calculate the probability of the event as: r1 = q*P11 + (1-q)*P10 and she will therefore bid r1, (If x1=0 she will bid r0=q*P01+(1-q)*P00)
Next turn • P2 doesn’t know what x1 is but he can infer it from the bid P1 made, i.e., r0 or r1. • P2 infers the x1 and knows x2 so he can post the best estimate of the conditional provability of E. • P2 will bid p00 or p01 or p10 or p11. • No one else will want to move.
Good market • All information will aggregate in just 2 steps. • Both players would make profit in expectation in the market.
Bluffing • Same thing but player 1 decides to bluff. Which means he sees x1=1 but moves to r0. • P2 infers P1 saw x1=0 and infers the probability knowing x2. (p00 or p01) • P1 sees the choice of P2, learns x2 and know x1 and moves to the best probability.
Bluff or tell the truth? • P1 lies because she has greater profit when doing so vs. playing myopically. • If P1 will benefit more from telling the truth, P2 have no reason to lie since P1 will not play again. • So bluffing can start only if P1 calculates that bluffing is better than telling the truth.
Bluffing by Player 1 (P2 is myopic) • If bluffing is more profitable then P1 will bluff with some probability s. P2 can analyze P1’s profit in different scenarios and decide that P1 is bluffing. • If P2 knows, his best response based on s, X2 and the price r0 or r1 has published can be calculated and he too can decide if to use it or to bluff.
Bluffing is Profitable • Let’s prove that player 1 has incentive to bluff which on information market with logarithmic scoring rule.
General Informativeness condition • We limit ourselves to the case in which every agent has something to tell us about the world, no matter what the other agents tell us. • Therefore we have to learn what each agent knows to make the best bid.
Let’s prove • 2 players, 2 signals. • Event prob. conditioned upon P1 seeing i and P2 seeing j is Pij • Player 2 has prob. q to see 1. • Player’s 1 myopic bids are: r1 = q p11 + (1-q)p10 r0 = q p01 + (1-q)p00
Equilibrium strategy profile • Each player’s strategy is sequentially rational , which meant that each player wants to maximize his own profit in expectation from making trades in the market, given all information that he knows on the event at each point of the game.
Proof • Assume Ru != R1. • Whenever player 1 will play Ru, player 2 will deduce player 1 saw 1 and will profit the remaining surplus. • So, Player one will always earn profit from only the first move, but by definition of myopic optimality R1 will yield better profit!
Two strategies • Myopic: Ps -> R1 • Bluff: Ps->R1->R0 • Ps->R1 cancels out. • Analyze the profit or loss from the movie from R1->R0
Weak PBE strategy porfile • 1) Unique case of general Equilibrium strategy profile. • 2) Strategies are sequentially rational given their beliefs. • 3) Updating the players beliefs is base on using Bayes’s rule given the strategies.
Proof • Let (S1,S2) be a weak PBE strategy. • Suppose S1 requires P1 to follow myopic strategy in the first round. • By lemma 1, P1 will have to bid r1 or r0. (when P1 sees x1=1 or x1=0 accordingly) • P2 will take into account and move to the optimal point. (p00, p01, p10, p11)
Proof • Consider a deviation from this strategy in which P1 bluffs and corrects P2’s move at the end. • Lemma 2 shows the expected additional score increase if P1 bluffed:
Proof • Inequality is strict when q!=0,1 and p10 != P00,P11 • Thus, bluffing will be strictly profitable deviation under this thereom. • Hence, myopic strategy of P1 cannot be part of an equilibrium profile.
No promised convergence • We showed that it is always profitable for the player to bluff and not play deterministically. • There are 2 cases: • Case 1: player 1 plays some strategy regardless of x1, In this case P2 learnt nothing about x1 and the theorem always holds after round 1.
No promised convergence • P1 plays mixed strategies and moves to Ru which with prob. T when x1=1 and prob. T` when x1=0 when T and T’ are non zero. so P2 can not infer exactly what is x1. so P2 assigns some prob. K to x1 = 1. (K != 0,1) The conditions in theorem 3 holds so P2 has also incentive to bluff. Thus the price cannot converge with certainty after N finite rounds!
Generalize the results • Lets look on m players with n signals.
Player 1 move • Player’s 1 myopic optimal moves: • Player’s 1 decides to bluff:
Homework • Prove claim 5.
Note about convergence • As in the 2 players model, convergence is not guaranteed, because bluffing is better in each round.
Fight the bluff • We want to cancel the incentive to bluff. • Let’s reduce the price paid for future trades! • Maybe even cause traders to take the myopic strategy. • New Scoring Rule:
Truth or lie? • The myopic strategies hold, since every round we multiply by const. • But this scoring rule is better on non myopic strategies. • Di quantifies the degree of aggregation in the prediction market.
Note on delta • Choosing small delta will speed up convergence to real price because the benefit from each additional trade will be less. • However too rapid drop will cause traders not to participate because of too small price.
