Decision Theory

Lecture 13

1. Riskiness measures and gambling wealth Riskiness measures – the idea and description

• Aumann, Serrano (2008) – economic index of riskiness• Foster, Hart (2009) – operational measure of riskiness

Buying and selling price for a lottery and the connection to riskiness measures• Lewandowski (2010)

Two problems resolved by gambling wealtha) Rabin (2000) paradoxb) Buying/selling price gap (WTA/WTP disparity)

The idea

Quest for operational meaning

• Log is „the first” function going from above that assigns – infinity to bankruptcy

Example

f(W)=Elog(W+x)-log(W)

500 550 600 650 700 750 800 850 900

-0.9

-0.7

-0.5

-0.3

-0.1

0.1

0.3Max loss

W<R(x) W>R(x)

W*=R(x)

L10 0,35 1 0,35 0Cost 0,25 -600 0,65 0,25 0,1625 -600

0,5 600 0,5 0,325 6000,25 1800 0,25 0,1625 1800

a m b f(a) f(m) f(b)600 900 1200 -7,80811128 0,255413 0,258518600 750 900 -7,80811128 0,197478 0,255413600 675 750 -7,80811128 0,093509 0,197478600 637,5 675 -7,80811128 -0,04136 0,093509637,5 656,25 675 -0,04136275 0,040451 0,093509637,5 646,875 656,25 -0,04136275 0,004558 0,040451637,5 642,1875 646,875 -0,04136275 -0,01687 0,004558

642,1875 644,53125 646,875 -0,01687236 -0,00582 0,004558644,5313 645,703125 646,875 -0,00581623 -0,00055 0,004558645,7031 646,289063 646,875 -0,00054848 0,002024 0,004558645,7031 645,996094 646,2891 -0,00054848 0,000743 0,002024645,7031 645,849609 645,9961 -0,00054848 9,85E-05 0,000743645,7031 645,776367 645,8496 -0,00054848 -0,00022 9,85E-05645,7764 645,812988 645,8496 -0,0002247 -6,3E-05 9,85E-05645,813 645,831299 645,8496 -6,3047E-05 1,77E-05 9,85E-05645,813 645,822144 645,8313 -6,3047E-05 -2,3E-05 1,77E-05645,8221 645,826721 645,8313 -2,2658E-05 -2,5E-06 1,77E-05645,8267 645,82901 645,8313 -2,4673E-06 7,63E-06 1,77E-05645,8267 645,827866 645,829 -2,4673E-06 2,58E-06 7,63E-06645,8267 645,827293 645,8279 -2,4673E-06 5,64E-08 2,58E-06645,8267 645,827007 645,8273 -2,4673E-06 -1,2E-06 5,64E-08645,827 645,82715 645,8273 -1,2054E-06 -5,7E-07 5,64E-08645,8272 645,827222 645,8273 -5,7449E-07 -2,6E-07 5,64E-08645,8272 645,827258 645,8273 -2,5904E-07 -1E-07 5,64E-08645,8273 645,827276 645,8273 -1,0131E-07 -2,2E-08 5,64E-08645,8273 645,827284 645,8273 -2,2442E-08 1,7E-08 5,64E-08645,8273 645,82728 645,8273 -2,2442E-08 -2,7E-09 1,7E-08

Lottery dillution