1. 1 Learning and discovery Simon Grant Rice University & John Quiggin Risk and Sustainable Management Group Schools of Economics and Political Science, University of Queensland

2. 2 Web sites RSMG http://www.uq.edu.au/economics/rsmg/ind ex.htm Quiggin http://www.uq.edu.au/economics/johnquig gin WebLog http://johnquiggin.com

3. 3 Learning and discovery Every solution of a problem raises new unsolved problems; the more so the deeper the original problem and the bolder its solution. The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance. For this, indeed, is the main source of our ignorance - the fact that our knowledge can only be finite, while our ignorance must necessarily be infinite - Popper, 1963

4. 4 Unforeseen contingencies Unknown unknowns (Rumsfeld) Discovery in scientific research Problems of decision theory Generalizations of EU within state-act framework Ambiguity and uncertainty Important contribution but don’t deal with crucial problem State space cannot be an exhaustive description

5. 5 Aims of this paper Provide a formal representation of the process of discovery Determine conditions under which standard Bayesian learning theory is applicable Consider implications of continuing discovery for decision theory

6. 6 Research and discovery Example of electron beam experiment Researcher decides whether to undertake experiment to test atomic hypothesis Act of Nature may produce unanticipated possibility Gamma ray emission

7. 7 Decision Tree for Electron Beam Experiment without Unconsidered Possibility of Gamma Ray Emission

8. 8 Tree for Electron Beam Experiment that Includes Unconsidered Possibility of Gamma Ray Emission

9. 9 Nodes and trees Tree structure Nodes, occurrences, histories, events, instants, Decision nodes represent acts of the individual or nature

10. 10 Propositions Sentences in a formal language Bounded rationality means that not all sentences are expressible Decision propositions correspond to nodes or events Modal logic of tense derived from tree structure

11. 11 Knowledge and modal logic Modal logic of knowledge Individual knows proposition p at node n if p is true at all nodes considered possible at n Derived operators for “knowing whether p”, and “considers p”

12. 12 The tree structure Lattice of trees More refined tree implies larger set of expressible propositions Maximal tree represents external or unboundedly rational viewpoint A subjective tree associated with each node in the maximal tree

13. 13 Propositions and the tree structure A proposition is expressible in a given tree structure if the set of nodes at which it is true in the maximal structure corresponds to a set of nodes in the given tree structure The more refined the tree structure, the larger the set of expressible propositions

14. 14 A Coarsening of the Tree in Figure 2

15. 15 Lattice construction Mappings from more refined to less refined trees, and inverse correspondences These induce awareness operators Need to show that such mappings commute

16. 16 Direct (and Indirect) Mapping from tree  ’’ to coarser tree  (via  ’).

17. 17 Commuting Information Correspondences for tree 

18. 18 Dynamics of learning and discovery Move from a node to its successor yields new, more refined subjective tree structure Newly expressible propositions are said to be discovered World is characterised by continuing discovery

19. 19 Awareness of unconsidered propositions The crucial innovation in this paper We use an existence operator to generate statements of the form ‘There exist propositions p that I have not considered’ Hence, individuals can reason about future discoveries

20. 20 Research example Investigator can entertain proposition ‘If I undertake action A, I will discover new possibilities’ This is more plausible if action A is ‘conduct experiment’ than if action A is ‘do not conduct experiment’

21. 21 Criminal investigation ‘Person A is a suspect’ There may exist evidence that, if discovered would imply guilt of Person A The nature of this evidence may not be known

22. 22 Contracting under ambiguity (Grant, Kline & Quiggin) Risk sharing contract A pays if card is black, B pays if it is white Unconsidered ‘grey area’ leads to a dispute Contracting may or may not be optimal

23. 23 Discovery and knowledge Proposition ‘I will be more aware in the future than I am now’ True if world is characterised by continuing discovery Propositions of this kind can not be known (in modal-logic sense) to be true

24. 24 Developments Probabilities Consistency conditions for Bayesian learning Ambiguity and multiple priors

25. 25 Potential applications Theory of research and discovery Precautionary principle Entrepreneurship

26. 26 Concluding comments The more we know, the more we discover our own ignorance Analysis here both formalises and exemplifies this point Given a formal way of describing discovery, can recognise great gaps in our understanding of this and related processes