Do Markets Favor Agents Able to Make Accurate Predictions?

1. Do Markets Favor Agents Able to Make Accurate Predictions? Alvaro Sandroni Reporter:Lena Huang

2. Introduction • A Model of Reinvestment • Endogenous Investment and Savings –Examples • A Model of Endogenous Investment and Savings • Basic Concepts • Predictions and Survival • Convergence to Rational Expectations • Conclusion

3. Introduction (1) • A long-standing theory in economics is that agents who do not predict as accurately as others are driven out of the market, and it underlies the efficient-markets hypothesis and the use of rational expectations equilibrium as a solution concept because it implies that asset prices will eventually reflect the beliefs of agents making accurate predictions. • However, under certain conditions the agents who have accumulated more wealth are also those who have made the worst prediction.Blume and Easley (1992) is an example.

4. Introduction (2) • Blume and Easley (1992) show that if agents’ have the same savings rule, those who maximize the expected logarithm of next period’s outcomes will eventually hold all wealth.However, if no agent adopts this rule then the most prosperous are not necessarily those who make the most accurate predictions. • Agents with incorrect beliefs, but equally averse to risk, may choose an investment rule closer to the MEL rule, and so eventually accumulates more wealth than the agent with correct beliefs.

5. Introduction (3) • The recent literature casts serious doubt on the theory that agents with incorrect beliefs will be driven out of the market by those with correct beliefs.This paper seeks to resurrect this intuitive theory. • The main difference is that,in the recent literature, savings are exogenously fixed and agents’ choices are solely restricted to investment decisions.In this model, I assume that agents make savings and investments decisions that fully maximize expected discounted utility.

6. Introduction (4) • In this paper, I show that if markets are dynamically complete then, among agents who have the same intertemporal discount factor (but not necessarily the same degree of risk-aversion), the most prosperous will be those making accurate predictions, and convergence to rational expectations obtains. • This holds even though agents with identical beliefs, but different utility functions (i.e., diverse preference over risk), may choose different savings and portfolios, and therefore, the relative wealth of agents with different preferences over risk is a random variable.

7. Introduction (5) • The intuition is that if agents’ choices are restricted to investment decisions, then they may optimally choose to allocate small amounts of wealth to events they believe likely to occur. • However, if agents can maximize over both savings and investment decisions, then I will show that although they may not maximize wealth accumulation, they will still allocate relatively more wealth to future events they believe more likely to occur;therefore,agents who eventually make accurate predictions survive in the market.

8. Introduction (6) • I examine who survives in the market without the assumptions that agents have identical discount factors and some make accurate predictions. • I show that any agent who has strictly smaller entropy than another agent is driven out of the market. • This result is particularly appealing because agents’entropy depends only on exogenous parameters.Therefore, it is possible to compute it without solving for equilibrium.

9. A Model of Reinvestment (1) • This model (following Blume and Easley(1992)) show that if savings are exogenously fixed, then agents with correct expectations may accumulate less wealth than agents with incorrect expectations. • The argument is completed by observing that agents with incorrect beliefs may choose an investment strategy closer to the MEL rule (which maximizes the growth rate of wealth) than what agents with correct beliefs choose.

10. A Model of Reinvestment (2) Some notations~ - N+ : N∪{0} -The set of states of nature is given by T≡{1, …,L}, L N. -Tt, t N ∪{∞}, be the t-Cartesian product of T. -For every finite history , a cylinder with base on stis the set C(st) = of all finite histories whose t initial elements coincide with st. - :the σ-algebra consisting of all finite unions of cylinders with base on Tt. - 0 ≡∪ ; hence, is the smallest σ-algebra that contains 0 .

11. A Model of Reinvestment (3) • The true stochastic process of states of nature is given by an arbitrary probability measure P defined on (T∞, ). • Given a t-history , let be the posterior probabilities of P, defined by : • Let EP and be the expectation operators associated with P and , respectively. Where is the set of all paths such that s=(st , ), .

12. A Model of Reinvestment (4) • A decision maker has initial wealth w0=1.At period t, she chooses a portfolio among M assets such that . • An investment strategy is a sequence of portfolios ,and gross returns of assets are given by the positive -measure functions • If consumption is a fixed portion,1-δ,of wealth, then agents who invest under strategy a accumulate wealth . • The entropy of a portfolio at is

13. A Model of Reinvestment (5) • Proposition 1: Let α1 and α2 be investment strategies such that and are uniformly bounded away from zero and infinity.If there is such that, for all , P-a.s., then . • Proof:Let , .By the law of iterated expectations, .By the law of large numbers for uncorrelated random variables,P-a.s., By assumption,. Hence,

14. A Model of Reinvestment (6) • The MEL rule is the myopic investment strategy a* defined by . • A corollary of Proposition 1 is that agents who adopt the MEL rule eventually obtain greater wealth than the others. • Blume and Easley (1992) extended this result to an equilibrium model, and show that only agents investing under MEL rule survive. But if no one have a log utility function, agents with the highest entropy (hence, accumulate most wealth and survive) are not necessarily those making correct predictions.

15. A Model of Reinvestment (7) • The key observation is that agents having correct beliefs and maximizing expected utility may not optimally invest under the MEL rule and maximize wealth accumulation, because of their preference over risk. • Agents with incorrect beliefs, but equally averse to risk, may choose an investment rule closer to the MEL rule, and hold portfolios with higher entropy, which eventually result in more wealth. • Under the assumption of savings being exogenously fixed, similar examples results can be constructed in many frameworks under fairly general conditions.

16. Endogenous Investment and Savings-Examples • In contrast with the above results, we will show that if savings are endogenous then agents making inaccurate predictions will necessarily be driven out of the market. • This section illustrate some of these ideas in simple examples of dynamically complete market economies.

17. Example 1 of Endogenous Investment and Savings (1) • Assumptions~ -Two long-lived agents, 1 and 2,two long-lived trees, 1 and 2, two states of nature, h and l, and one consumption good c. -Tree 1 gives 0 units of consumption in state h and 1 unit in state l. Tree 2 gives 2 units of consumption in state h and 0 unit in state l . -The probability of state h is 0.5 in every period. -Agent 1 has correct beliefs.Agent2 believes that the probability of state h is 0.5+ε.Let Pi be the probability measure associated with the agent i’s beliefs.

18. Example 1 of Endogenous Investment and Savings (2) • At period t, the price of tree m is given by pm,t, agent i’s consumption and share holdings of tree m are given by Let be agent i’s wealth, • Agents’ maximizing problem is : (State l ) (State h) Total dividends at t

19. Example 1 of Endogenous Investment and Savings (3) (A) • From two agents’ f.o.c, where n is # of times that state h occurs until t. • (A) shows that the ratio of probabilities times MU of consumption is constant. • By the law of large numbers, if t is large, n should be close to t/2.Therefore, • By (A) and (B), (B) (C) (D)

20. Example 1 of Endogenous Investment and Savings (4) • Equation (D) shows that agent 2 allocates vanishing consumption in some paths believing they will not occur, and share prices converge to REE in which agent 1 is alone in the economy. • However,in both states, agent 1’s portfolio are further away from MEL rule than agent 2’s portfolio.This is possible because of the difference in savings behavior. • Agent 2’s compounded saving ratio(the product of agent 2’s savings ratio in all period) is eventually an arbitrarily small fraction of agent 1’s.This fraction will eventually be small enough to compensate for difference in portfolio selection, making the relative wealth of agent 2 small. The expected log of agent 1’s saving ratio: is greater than the expected log of agent 2’s saving ratio (log(0.5))

21. Example 2 of Endogenous Investment & Savings • Assume thatε=0, i.e., the beliefs of both agent are identical. • There will be no speculative trade and agent may achieve an efficient allocation by trading just once. • Agent 1 always holds 2/3 of tree 1 and ¾ of tree 2; Agent 2 always holds 1/3 of tree 1 and ¼ of tree 2, and consuming dividends given by these shares. • Agents’ wealth and consumption depend only on the current state. • Both agents survive, although they do not choose similar savings and portfolios. • In both cases, share prices are different, but eventually close to a REE.

22. A Model of Endogenous Investment & Savings-The general case • Assumptions of the model : -I long-lived agents, M long-lived trees, L states of nature, h and l, and one consumption good c. -Agents are born with shares of the trees and receive no other endowments. -Dividends: ; e=(et , ) -Share price: ; p=(pt , ) -Agent i’s consumption: -Share holding: ; *Hence, agent i’s wealth is given by

23. A Model of Endogenous Investment And Savings (1) • The model of dynamically complete markets: -Let P & Pi represent the true probability measure & agent i’s beliefs respectively. -Let Htbe the -measurable function defined by -The markets are dynamically complete.That is L=M, and the rank of Ht(s) is L.So agents may transfer wealth across states of nature by trading the existing assets.

24. A Model of Endogenous Investment And Savings (2) • Agents’ maximizing problem is • Where uiis a strictly increasing, strictly concave, continuously differentiable utility function that satisfies the Inada conditions.

25. Basic Concepts • Survival • The Accuracy of Agents’ Predictions • Entropy

26. Survival • Def 1:Agent i is driven out of the market on a path if agent i’s wealth, , converges to zero as t goes to infinity. Agent i survives on a path if he is not driven out of the market on s. • I focus on accumulation of wealth as main criteria to define survival because only agents with positive wealth influence prices.

27. The Accuracy of Agents’ Predictions (1) • The difference between two probability measures in the sup-norm, is given by • The difference between , in the dl-metric, is given by . Two probability measure are close, in the sup-norm, if they assign “similar” probability to all events. Two probability measure are close, in the dl-metric, if they assign “similar” probability to events within l-periods.

28. The Accuracy of Agents’ Predictions (2) • Def 2: Agent i eventually makes accurate predictions on a path , s=(st,…), if • Def 3:Agent i eventually makes accurate next period predictions on a path , s=(st,…), if . • Def of ‘merge’:Agent i’s beliefs (weakly) merge with the truth if, P-a.s., agent i eventually makes accurate (next period or l-period) predictions. Merging implies weak merging, but not conversely.

29. The Accuracy of Agents’ Predictions (3) • Def 4: :Agent i eventually makes inaccurate next period predictions on a path , s=(st,…), if there isε>0 such that • Clearly, an agent who does not eventually make accurate next period predictions need not always make inaccurate next period predictions. • Def 5:Some agents eventually make accurate (next period) predictions if, P-a.s., in every path , there exists at least one agent who eventually makes accurate (next period) predictions on s. (Not necessarily the same agents on different paths)

30. Entropy (1) • Let , then the probability of the states of nature at period t, Qt, is defined as • In particular, are the true probabilities and agent i’s beliefs over states of nature at period t, given past data, respectively. • Def 6 :The entropy of agent i’s beliefs at period t, , is given by • and it is zero iff agent i’s belief and the true probabilities over states of nature in the next period are identical. Write on the blackboard

31. Entropy (2) • Def 7:The entropy of agent i, is given by: • Hence, the entropy of an agent does not depend on the characteristic of the other agents. • Def 8:The ratio of beliefs and true probabilities over states of nature in next period uniformly bounded away from zero and infinity if there exists u>0 and U<∞ such that i.e.Pt≧ε＞0 Pti≧ε＞0

32. Predictions and Survival • Main Results -proposition 2 -proposition 3 -proposition 4 -proposition 5 • Proofs and Intuition -Basic Results -Results in Probability Theory -Proof of Proposition 2-5

33. Main Results-Proposition 2 (1) • Proposition 2:Assume that all agents have the same intertemporal discount factor and some agents eventually make accurate predictions.Then in every equilibrium, P-a.s.: 1.Any agent who does not eventually make accurate predictions on a path is driven out of the market on the path s. 2.Any agent who eventually makes accurate predictions on a path survives on s. • Agents with diverse preferences over risk survive with probability 1 if their beliefs merge with the truth.This holds although the relative wealth of agents is a random variable because they may choose different savings and portfolios.

34. Main Results -Proposition 2 (2) • Proposition 2 is surprisingly strong. For example, agent 1’s belief merge with the truth, but agent 2’s weakly merge with the truth.The differences in belief have vanishingly small impact on agents’ savings and investment decisions. However, by proposition 2, agent 2 is driven out of the market with probability 1. • A corollary of proposition 2: Assume that all agents have the same intertemporal discount factor. In every equilibrium, agent i survives Pi-a.s. Consider the case P=Pi.Then agent i’s beliefs are exactly correct.By proposition 2, he survives Pi-a.s.

35. Main Results -Proposition 2 (3) • Intuition -Agents who maximize expected discounted utility functions allocate relatively more wealth to paths they believe more plausible than to paths they believe less plausible. -Thus, agents who eventually make accurate predictions allocate large amounts of wealth to paths that have, in fact, high probability and, hence, survive.

36. Main Results -Proposition 3 (1) • In the following proposition, I relax the assumptions that all agents have the same discount factor and that some agents eventually make accurate predictions. • Proposition 3:Assume that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from zero and infinity. In every equilibrium, P-a.s., if the entropy of agent i is strictly smaller than the entropy of agent j on a path , , then agent i is driven out of the market on s.(remind entropy)

37. Main Results -Proposition 3 (2) • Agents’ entropy is a function of exogenous parameters, so there is no need to solve for equilibrium to compute it.Moreover, the entropy of an agent does not depend on preferences over risk, dividends in each state of nature, and beliefs and discount factors of the other agents. • Agents whose entropy is not smaller than any others’ do not necessarily survive.For example, agent 1 has correct beliefs; agent 2’s beliefs weakly merge with the truth.The average entropy of agent 2’s beliefs is zero.So, their entropy is identical.However, by proposition 2, agent 2 is driven out of the market.

38. Main Results -Proposition 4 • Because that the entropy of an agent who always makes inaccurate next period predictions is strictly smaller than that of an agent who makes accurate next period predictions, if they have the same discount factor.(Show in next section)Proposition 4 follows from this observation and Proposition 3. • Proposition 4:Assume that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and ∞; that all agents have the same intertemporal discount factor;and that some agents eventually make accurate next period predictions.In every equilibrium, P-a.s., if agent i always makes inaccurate next period predictions on a path ,then agent i is driven out of the market on s. (However)

39. Main Results -Proposition 5 • Proposition 5:Under the same assumptions of proposition 4, in every equilibrium, P-a.s., if there exists andε>0 such that on a path , s=(st,…), then agent i is driven out of the market on s. • An open question is whether proposition 3~5 are true without the assumption that the the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and infinity. Stronger than proposition 4

40. Proofs and Intuition-Basic Results (1) • Agents’ f.o.c of the maximization problem imply that, • Lemma 1:In every equilibrium, for any path and agent , Lemma 2:Fix an agent and a path In every equilibrium, if there exists an agent such that then agent i is driven out of the market on s. Moreover, if for all agents there exists ε>0 such that then agent i survives on s. ( * )

41. Proofs and Intuition-Basic Results (2) • Lemma 2 implies that, almost surely, if agent i believes that a path s is much less likely to occur than agent j does, and they have the same discount factor, then agent i allocates much less wealth on s than agent j does and, hence, is driven out of the market on s. • Lemma 2 can be used to determine who survives in simple examples. Here is an example in which an agent whose beliefs weakly merge with the truth is driven out of the market although no other agent eventually makes accurate next period predictions.

42. Proofs and Intuition-The example (1) • The true probability of state a is 1, and agent 1 believes that a will occur next period with probability , so agent 1 eventually makes accurate next period predictions.At period 0, agent 1 believes that state a will always occur until period t with probability . • Let t(k), , be the smallest natural number such that .At periods t(k), agent 2 believes that state a will occur next period with probability 0.5.In all other periods, agent 2 believes that state a will occur next period with probability 1.Agent 2 does not eventually make accurate next period predictions because, infinitely often, agent 2 believes that state a will occur next period with probability 0.5.

43. Proofs and Intuition-The example (2) • At period t, t(k)<t≦t(k+1), .Hence, By Lemma 2, agent 1 is driven out of the market. • If there were another agent, agent 3, who eventually makes accurate predictions, then By definition, .Thus, by Lemma 2, agent 2 is driven out of the market. ＊In the example above, the true distribution is deterministic.The results of the next section deal with arbitrary stochastic processes.

44. Proofs and Intuition -Results in Probability Theory (1) • Lemma 3:For every agent , P-a.s., .Moreover, P-a.s., agent i eventually makes accurate predictions on a path iff . • Lemma 4:Assume that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and ∞. Agent i eventually makes accurate next period predictions on a path iff . Agent i eventually makes accurate next period predictions on a path s iff there existsδ>0 such that .

45. Proofs and Intuition -Results in Probability Theory (2) • Lemma 4 is not true without assumption that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and∞. • For example, there are two states of nature a & b. At period t, agent i believes that state a will occur next period with probability .The true probability is 1/t.Agent i weaklymerge with the truth.However,

46. Proofs and Intuition –Proof of Proposition 2-5 • Proof of Proposition 2 • Proof of Proposition 3 • Proof of Proposition 4 • Proof of Proposition 5

47. Proof of Proposition 2 • If there is an agent j who eventually makes accurate predictions on s, by Lemma 3, .If agent i does not eventually make accurate predictions, by Lemma 3 .Thus, By Lemma 2 agent i is driven out of the market. • If agent i eventually makes accurate predictions, by Lemma 3, .Moreover for for all agents j, on s.Hence, on s for all j. By Lemma 2, agent i survives on s.

48. Proof of Proposition 3 (1) • Let , and let . By the law of iterated expectations, .By the law of large number for uncorrelated random variables, • If , by definition, . Hence,

49. Proof of Proposition 3 (2) • By definition, • By Lemma 2, agent i is driven out of the market on s.

50. Proof of Proposition 4 • Assume that agent i always makes inaccurate next period predictions on a path s, that agent j eventually makes accurate next period predictions, and they have the same discount factorβ, by Lemma 4, and ≦δ＜0 By proposition 3, agent i is driven out of the market.