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"A million times a million is a billion" should be "A million times a million is a trillion"
There's a standard explanation in game theory as to why wars or fought (or lawsuits are brought to trial, etc.), even when everyone would be better off with a negotiated settlement that avoids expensive conflict. Even assuming that it's possible to enforce an agreement, there's still a problem. In order to induce militarily strong parties (in terms of capability or willingness to fight) to accept the agreement rather than fight, the agreement must be appropriately tilted in their favor. If there is asymmetric information about the relative military strength of the parties, then in equilibrium, wars must be fought with positive probability.
There's an error in your 1% of a billion dollars example.
You write one million dollars x 100, which is 10% of one billion dollars, not 1%
It would only take 9 more blocks like this to get back to the one billion dollars.
Something very close to this is done with fleet cards (cards given by long-haul trucking companies to their drivers to pay for gas and other expenses). At the high-end, these cards capture a great deal of data, comparable to a receipt. (Not exactly equivalent -- it's more detailed then a typical receipt for some data fields, but doesn't include everything that can be put on a receipt). It's expensive to implement and maintain these sorts of data capture systems, since a receipt is very flexible about the data it can contain. As a result, more broad implementation doesn't make financial sense. The fraud gains are minimal. It's worth doing in the fleet card segment because it helps with a) expense tracking, b) enforcing agreements between the large trucking companies and the chain gas stations regarding fuel discounts, and c) optimizing the choice of gas station (trading off the route efficiency against lower fuel prices). When I last consulted in the industry, this kind of data capture wasn't even a close to being profitable for general-purpose credit cards. (It's been ~5 years so my knowledge may be a bit out of date.) If you scaled back the requirements, it might be feasible.
TLDR: The paradox goes away if you make price endogenous, i.e., it only occurs because your assumption about the value growth over time that is inconsistent with the profit flows.
The paradox stems from the fact that you've made inconsistent assumptions: that the value of the company increases linearly over time, and that the company never generates a flow of profits (i.e., the only value comes from the sale). If profits are zero, the equilibrium price is constant at zero, and investors are indifferent between holding the company and selling it at any point in time. More generally, if the company has some potential for profits (which can be modeled as a flow of profits per unit of time, or as a hazard rate of getting an instantaneous lump sum of profits), the equilibrium price will be set so that the marginal investor is indifferent between holding and selling.
I have a tongue-in-check resolution to the Schrodinger cat variant: if his goal is to set a new world record, he should open the box immediately after the old world record. More seriously, to resolve the paradox, you need to be more explicit about his utility function: how does the value he obtains increase with the amount by which he exceeds the old record? Depending on your choice of utility function, you may or may not have a paradox, and it may or may not be equivalent to the St. Petersburg paradox.
Two studies that have found some evidence that the timing of death responds to incentives created by changes in estate taxes:
1. From the USA: Kopczuk and Slemrod (2003), https://pdfs.semanticscholar.org/afd2/7f878ba71c9ec28258407a1e9b34150cd9d0.pdf
2. From Sweden: Eliason and Ohlsson (2013), http://www.diva-portal.org/smash/get/diva2%3A457378/FULLTEXT01.pdf.
These are far from definitive, but are definitely suggestive.