assuming nothing else, like a time dependent random number generator, could have affected the order of assigned numbers here). However, what if the actual drawing process is influenced by your selection $^{14}$? In such a case the outcome is conditional on your choice. Knowing that you were right/wrong with one choice does then not mean that you would have not been right/wrong with another choice. In other words, the lottery winner could still have been win- ning (though probably with a low chance) if he would not have swapped. More importantly, the other per- son who would have gotten the numbers now first probably would have not won as the outcome was conditional on them swapping. To extract the essence consider the following: a game master preselects (lets say by some “random method” like throwing a dice) two possible outcomes (0 or 1) for A and B separately. You are now asked to guess an answer (0 or 1). If you say 0 the game master reveals to you the outcome drawn for A, while if you say 1 he will reveal outcome B. Hence, having been right with your guess does not tell you whether you would have been right or wrong would you have chosen the other answer. 15 These situations arise if the observer is embed- ded in the system he is analysing and his interac- tions cause a meaningful disturbance in the system. Chaotic systems are extremely sensitive to variations in initial conditions and hence one can imagine that even just the presence of an observer (gravitational interaction) can change an outcome. We believe that such issues are closely related to the concept of Free-Will. Knowing that your ‘act of As people usually subscribe to certain scenarios (“I’m a bull” or “I’m a bear”) and do not allow for probabilistic weightings, such situations appear real- istic. Interesting phenomena should arise if there are dis- tinct groups of market participants, which assign very different valuations to an asset. As long as there are enough buyers and sellers in the group attaching the highest valuation, the situation can be stable. How- ever, as the size of this group shrinks (for instance due to new information) sellers might not find buyers in this group and the price can collapse. One could see this as a kind of “phase-transition” from one stable state into another. Hence for an observer inside the chaotic system the system is unpredictable and dependent on his deci- sions — he perceives Free-Will. prediction’ is influencing the future itself makes it im- possible for you to predict the future. Hence even a deterministic system will be unpredictable from within if the act of prediction cannot be performed without interfering with the system. E Segmented Markets and Phase Transition References [1] Jaynes E T, Bretthorst L G (2003), Probability Theory: The Logic of Science , Cambridge Uni- versity Press,Cambridge [2] Kirchner U (2009), ‘Market Implied Probabil- ity Distributions and Bayesian Skew Estimation’, arXiv 0911.0805[q-fin.PR] (Online at http://arxiv.org/abs/0911.0805 ) [3] Haug E G, Taleb N N (2009), ‘Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula (fifth version)’, SSRN 1012075 (Online at http://ssrn.com/abstract=1012075 ) [4] Hull J C (2005), Options, Futures and Other Derivatives , Prentice-Hall, 11 15 A variation on this theme is a game where a game master choses an answer based on a probability distribution depen- dent on your choice. For example you choose between yes and no, and he selects the answer to be (a) 50% yes/50% no if you selected yes and (b) 20% yes/80% no if you selected no. Play- ing the game repeatedly you will now find that the distribution of outcomes depends on your bets. 14 We argue that due to the chaotic nature of our world this is a lot more likely than the reader may think. Some contem- plation shows that there are many instances in life where a tiny change in initial conditions has unpredictable effects. Let us only mention conception here, where the smallest change would lead to a different genetic make-up of a living being (de- termining gender etc.) — a huge factor in the parents further life.