May 17, 2013
May 17, 2013

Time frames & risk perception

Focus on the signal, not the noise

If World No. 1 Novak Djokovic was playing World No. 60 Paul-Henri Mathieu in a five set Grand Slam tournament, what respective odds would you expect for this game? How about Djokovic 1.72, Mathieu 2.40? Does that seem realistic?

If these odds appear a bit off, it’s probably because you typically wouldn’t give Djokovic a slim 58% chance of winning and the Frenchman a somewhat generous 42% chance. These are real odds, however, but just not for the match outcome.

Instead, these are the odds of winning each point, accurate for this pair’s encounter in the first round of 2013 Australian Open, which Djokovic went on to win (6-2, 6-4, 7-5).

Let’s break the numbers down. The game featured 163 points, of which Djokovic won 95 and Mathieu 68. From this we can produce a probability per point of 95/163 for Djokovic and 68/163 for his opponent. The actual odds for the entire match were entirely different at

Djokovic:   1.01Mathieu:   41.00

Both sets of odds are accurate, but are looking at the game from entirely different perspectives or “frames”: the narrow frame (one point) vs. the broad frame (the entire match).

People struggle to reconcile the differing probabilities of the two perspectives – even though they are for the same event – and this can have significant consequences, especially in live tennis betting.

Live betting effectively looks at the game point-by-point, and at the scale Mathieu truly does have a 42% success rate. This can lead bettors to overstate his overall chance, however, especially if they get caught up with the emotion of the crowd and the commentators.

People struggle to reconcile the differing probabilities of two perspectives

A 42% win percentage per point, extrapolated over all 163 points, however, translates to a chance of 41.00 for overall success as suggested by the odds (which suggests that if the pair played 41 times the Frenchman would win once).

Framing: a market for success

The phenomenon of framing bias has been documented by decision-making theorists and is seen where someone is averse to an isolated gamble, because subjective reason believes it to be risky despite the expected value being positive. A famous example is:

Heads loses you \$100, tails wins you \$200

The majority of people, displaying common risk aversion, reject the offer, focusing on the potential loss in the single toss scenario, even though the Expected Value, and therefore the probability of making money, is \$50:

Objective Expected Value = (0.5*200)-(0.5*100) = \$50

The academic reasoning for this is that people tend to feel the loss of \$1 twice as much as the gain of the same amount. Applying this factor of two for losses, the Expected Value for the single gamble to zero and hence it is rejected:

Loss Averse Expected Value = (0.5*200)-(0.5*100*2) = \$0

By expanding our framing to more than a one-off, however, we can get a better understanding of how much we’re likely to win or lose. Compare this slightly larger frame of two coin tosses:

As above, but with losses doubled due to loss aversion:25% lose \$400; 50% win \$100; 25% win \$400 – EV is \$50

Even with a heightened fear of loss, the Expected Value of two coin tosses is still more positive when viewed with a wider frame. Had you simply used the narrow-frame to analysis the outcome of the two tosses, the opportunity to benefit would have been missed.

The loss adverse approach to this bet tends to remain consistent when considering multiple tosses, but as the cumulative odds of losing diminish with aggregate gambles, loss aversion correspondingly diminishes.

If asked the same question based on a large number of tosses – a broader frame – people tend to be more comfortable with the gamble.

At 100 coin tosses, this bet (without the doubling) has an EV of \$5,000 with only a 1 in 2,300 chance of losing any money and a 1 in 62,000 chance of losing \$1,000. However, by rejecting the isolated bet in the first instance – the narrow frame – you miss out.

Probability of success at different frames
FramesProbability
1 year 93%
1 quarter 77%
1 month 67%
1 day 54%
1 hour 51.30%
1 minute 50.17%
1 second 50.02%%

Wider examples

The issues around framing are frequently observed in environments where frequency of data change is high, e.g. financial indexes. The more often you check the market – and therefore the narrower the frame you take – the more likely you are to see noise rather than signal.

Nassim Nicholas Taleb summarises it neatly in his book Fooled By Randomness by illustrating that a portfolio of stocks with a 15% return and 10% volatility returns surprisingly different chances of success at ever narrowing frames. If you were checking this portfolio every second the chance of success would be only 50.02% yet the broad frame – over a year – is 93%.

The lesson to learn from narrow framing is try to see gambles in the aggregate form, thereby avoiding missing out on what are actually favourable outcomes that your risk averse nature would otherwise intuitively reject.

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