Re Jeffrey-Bolker, the only system I studied in detail was Savage’s, but my impression is that the fix I applied to that system can be applied all the others that paint themselves into the corner of bounded utility, and with the same effect of removing that restriction. Do the Jeffrey-Bolker axioms either assume or imply bounded utility?
Having now read some expositions of the Jeffrey-Bolker theory, I can answer my own question.
The Jeffrey-Bolker axioms imply the finite utility of every prospect (to be technical, the Averaging axiom fails when there are infinite utilities), but the utility can be unbounded above and below. It cannot be infinite. In this it differs from Savage’s system.
For Savage’s axioms, unbounded utility implies the existence of gambles like St. Peterburg, of infinite utility, and all the rest of the menagerie of infinite games listed in this SEP article. From these a contradiction with Savage’s axioms can be found. Hence all models of Savage’s axioms have bounded utility.
In the Jeffrey-Bolker system, gambles cannot be constructed at will. The set of available gambles is built into the world that the agent faces. The agent is an observer: it cannot act upon the world, only have preferences about how the world is. None of the paradoxical games exist in a model of the Jeffrey-Bolker axioms. They do allow the existence of non-paradoxical infinite games, games such as Convergent St. Petersburg, which is St. Petersburg modified to have arithmetically instead of geometrically growing payouts. However, I note that one of Jeffrey’s verbal arguments against St. Petersburg — that no-one can offer the game because it requires them to be able to cover arbitrarily large payouts — applies equally to Convergent St. Petersburg.
Re Jeffrey-Bolker, the only system I studied in detail was Savage’s, but my impression is that the fix I applied to that system can be applied all the others that paint themselves into the corner of bounded utility, and with the same effect of removing that restriction. Do the Jeffrey-Bolker axioms either assume or imply bounded utility?
Having now read some expositions of the Jeffrey-Bolker theory, I can answer my own question.
The Jeffrey-Bolker axioms imply the finite utility of every prospect (to be technical, the Averaging axiom fails when there are infinite utilities), but the utility can be unbounded above and below. It cannot be infinite. In this it differs from Savage’s system.
For Savage’s axioms, unbounded utility implies the existence of gambles like St. Peterburg, of infinite utility, and all the rest of the menagerie of infinite games listed in this SEP article. From these a contradiction with Savage’s axioms can be found. Hence all models of Savage’s axioms have bounded utility.
In the Jeffrey-Bolker system, gambles cannot be constructed at will. The set of available gambles is built into the world that the agent faces. The agent is an observer: it cannot act upon the world, only have preferences about how the world is. None of the paradoxical games exist in a model of the Jeffrey-Bolker axioms. They do allow the existence of non-paradoxical infinite games, games such as Convergent St. Petersburg, which is St. Petersburg modified to have arithmetically instead of geometrically growing payouts. However, I note that one of Jeffrey’s verbal arguments against St. Petersburg — that no-one can offer the game because it requires them to be able to cover arbitrarily large payouts — applies equally to Convergent St. Petersburg.