There is an error in Table 1 that affects what we've written in draft 8 . First look at the certainty equivalents for the final wealth positions I've ploted for CRRA r=0.57 guys and CRRA for nearly risk neutral guys r=0.01 in the following two graphs. I have also added a CRRA r=0.77 type, mpore risk averse, for comparison. Basically for risk neutral types the problem we discuss in draft 8 in regard to force feeding doesn't arise (becasue risk neutral guys always go to corners on linear budget sets) , but it does when we relax the assumption of risk neutrality. I have made a suggested rewrite of page 3 and 4 text below the graphs . let me know what you think. Most of the commentary Glenn wrote on this matter still holds..but it is only relevant only when we allow for risk aversion . I'll have to have a look at the instructions for the subjects but i dont think there will be an issue (at worst something about net vs gross, )? and hopefully nothing about the interface
Glenn here is my reworking of the bookie spreadsheet
here is the crra57case (i have put the certainty equivalents for the optimal free range choices at the 9 house odds as grey dots); note that the certainty equivalents are all pretty close to $1 in the range 0.5 to 0.9, but there is still some stronger discrimnation going on in bettingbeaviour here -betting is only zero when house odds equal subjective beliefs)
for comparison purposes here is an even more risk averse CRRA r=0.77 ; the noticeable feature is that the force feeding certainty equivalents are much less than the certainty equivalents with free range optimal choices, on the order of 50 cents to several dollars for bets where at most $1 is at stake; this is not suprising since being forced to bet when u are more risk averse is going to create disutility; note that I haven't yet plotted any graphs concerning payoff dominace issues relevant to free range (the grey dots represent certainty equivalents from optimal choices...from which we have to infer grapohs all reflect optimal
[G&S&L: I have put suggested changes in red]Consider a subject that has revealed that they have a personal belief that A will occur with probability 3/4. Assume for the moment that the subject has to place a $1 bet with each bookie, as shown in Table 1 (JF comment : eliminate panel B from the table and rely on the graphs below) . This assumption is reflected in the fraction to bet being set to 1 in each row. Such a subject would bet on A for every bookie offering odds that corresponded to a lower probability of A winning than 3/4, and then switch over to bet on B for every bookie offering odds that corresponded to a higher probability of A winning than 3/4. (JF comment : when u think about it this last sentence says it all – a risk neutral guy will always go to the corners….)This betting pattern is shown in Table 1, and implies net earnings of $9 or -$1 with the first bookie, $4.00 or -$1 with the second bookie, and so on. The expected net earnings from each bookie can then be calculated using the subjective belief of 3/4 that the subject stated. Hence the expected net earnings from the first bookie are (3/4 *$9) + (1/4 * -$1) = $6.50, and so on for the other bookies. As shown in the second last column of Table 1 expected net earnings are always positive, implying that for a risk neutral person betting is always better in expectation than not betting., a fact which we return to in a moment. Overall expected income is given by gross expected income from bets laid with the bookies plus any stake not bet. Since all of the stake available for betting on each and every bookie is assumed to be bet in this case, total earnings are expected to be $14.20 + $9 endowment ($1 for each bookie) or $23.20. The calculations and comparisons in these tables is represented concisely in Figure X...: Put figure around here
Now consider a subject who is not risk neutral but risk-averse, holding subjective beliefs constant on A winning at 0.75. Assuming a CRRA subject with "typical" relative risk aversion coefficient of r=0.57, we recalculate the payoffs in certainty equivalent terms for the three actions the subject can take, bet all on A, bet all on B, or don't bet at all, in Figure Y Place Figure Y around here . It is clear from this graph that this risk averse subject will bet on A at all house probabilities (and associated odds) in the discrete range 0.1,0.2,......0.9 when she is forced to bet her $1 stake on either A or on B ,since the certainty equivalent of the risky $1 bet on A is always higher than the certainty equivalent for the risky $1 bet on B for these house probabilities. But the graph also makes it clear that these forced risky bets are worse than not betting at all whenever house probabilities are in the range 0.52 to 0.94 [guys, 52 to 94 is an eyeball approximation I can work out the exact numbers]. The ability not to bet, or at least not to bet very much, matters to this risk averse subject, even when the stake is only $1. The figure shows (grey dots) the certainty equivalents for optimal choices from the nine implied budget sets, one for each bookie, and these are closely approximated by the certainty equivalents from the no betting option in the range of house probabilities 0.5 to 0.9. As would be expected for a risk averse subject, and unlike the case for a risk neutral subject, when the house probabilities are close to the subjects' beliefs, not much betting will take place. [I'm not so sure that these last 2 sentences are right here...JF]
For comparison purposes
Glenn maybe u can rewrite the rest here - not muchreally changes except the
referenc to panel B...
The absence of betting in panel B is worth pausing over. In our elicitation
mechanism we are still able to recover the true belief of the subject, since
that was needed to identify which bookies were worth betting with. But if we
simply observed the bets of the subject, what would be infer? We could infer
that the subject believed that A would occur with probability greater than
0.4 and less than 0.9, which is not particularly informative.2 It is not a
wrong inference, since the true belief is 3/4 by assumption, but it is not
very precise. Moreover, the sample selection that occurs as one moves from
panel A to panel B is suggestive of the difficulty of making inferences about
true beliefs from market odds based solely on realized bets, since this subject
would not have had any impact on the market within this wide range.
An additional implication of allowing for the absence of betting concerns the
temporal
2 One could imagine a longer list of bookies, arrayed in finer increments of
odds, but this would not provide significant improvement on the possible inferences
from the observed bets alone.
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difference between the investment in the bet (today) and the expected payoff
(tomorrow). When subjects place a bet today they give up the stake: virtually
every betting house3 requires that bets be covered immediately with transfers
of funds. But the payout is defined in terms of future dollars. So any bettor
must discount the future payouts, conditional on outcome, to compare with the
current outlay. This implies that individual discount rates must play a role
in inferring individual beliefs from naturally occurring bets.4
There is also a compelling behavioral hypothesis motivating the option of allowing
subjects to bet or not bet: the hope that this will mitigate hypothetical bias
in responses. The evidence for this hypothesis comes from experiments in marketing
using so-called “stated choice experiments” to value private commodities
with different characteristics. Of course, to economists these are simply revealed
preference choices. For example, Lusk and Schroeder [2004] conduct a careful
test of hypothetical bias for the valuation of beef using these methods. They
consider 5 different types of steak, and vary the relative prices of each steak
type over 17 choices. For the subjects facing a real task, one of the 17 choices
was to be selected at random for implementation. Subjects also considered a “none
of these” option that allowed them not to purchase any steak. Each steak
type was a 12 ounce steak, and subjects were told that the baseline steak,
a "generic steak" with no label, had a market price of $6.07 at a
local supermarket. Each subject received a $40 endowment at the outset of the
experiment, making payment feasible for those in the real treatment. Applying
the statistical methods commonly used to analyze these data, they find significant
differences between
3 We exclude those houses interested in making exorbitantly priced loans and/or
backed by the Mafia, as any fan of The Sopranos can testify.
4 In betting markets one might simply assume that these factors, risk
attitudes and discount rates, cancel out on both sides of the market. We have
no sense of the veracity of this casual assumption. A lively debate on these
issues was initiated by Manski [2006], who formally challenged the claim that
equilibrium prices in prediction markets necessarily reflected the average
belief of traders. Responses from Gjerstad [2005] and Wolfers and Zitzewitz
[2005] suggest that the two may be relatively close for plausible environments.
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we not be?
My aim here is to sketch out the pattern of optimal choices for a variety of preference functions:
with parameters suggested by your estimations in other contexts. Basically I want to keep "close" to parametr settings in risk neutral eg.
I start with CRRA, r=0.57, beliefs on event A = 0.75 , beliefs on event B = 0.25 [this is as fgar as i got tonite]
First examine the Free Range idea - with $1 to spend on a particular house bet on A or on B with house probs {p,1-p}, or odds on A of 1/p and odds on B of 1/(1-p).
Here is the table of optimal contingent wealth holdings from a budget line p*x1+(1-p)*x2=1 as p ranges over the values in our experiment. I have placed a red line between the house odds where a "switch" occurs, ie the switch from having more contingent wealth on B than on A. At house probs below 0.75, ie at house odds above 1/0.75=$1.33 , we expect more final wealth on A than on B. All of these final wealth baskets are affordable at the given prices {p,1-p} and with a budget income of $1.
These final wealth allocations can also be described in terms of a division of 1$ wealth into fraction of 1$ spent on A . The graph illustrates this using the budget line corresponding to p=0.4 or odds on A of $2.50=1/0.4. The optimal final wealth choice has xA>xB ,so we can think of the decision process as retaining 16 cents for sure (the amount of wealth in the lower wealth state, xB) and spending the difference, $1-0.16=$0.84, on A. That is the fraction of wealth spent on betting is 84%, with the rest retained, and that 0.84 is spent on A at odds of $2.50 to buy a contingent commodity paying 0.84*2.50 =
lets use the precise numbers:
0.4*2.25799+0.6*0.161338=1 from the budget line
now add and subtract 0.4*0.161338 to get
0.4*2.25799-0.4*0.161338+0.4*0.161338+0.6*0.161338=1
which yields
0.4*[2.25799-0.161338]+0.161338=1, indicating that the budget of $1 is "allocated" to 16¢ for sure and spending of 84¢= 0.4*[2.25799-0.161338]
on a contingent commodity, wealth if A, in amount [2.25799-0.161338]=$2.10,approximately, which is the return on spending 84¢ at odds of 1/0.4=$2.50 .Now look at the force feeding option (at this particular budget line). The following graph shows the difference that the choice not to be can make .Well it actually is only schematic (but the certainty equivalents at the force fed choices are exact). We're going to get the same sort of phenomena as with risk neutrality , with a switching point under force feeding....but these are forced choices the subjects aren't going to like making. The table tells us the pattern of optimal choices and as house probs approach beliefs the subject is going to want to equalize contingent wealth...but can't under the forc feeding option.
john