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Articles :: More Healthcare IT Articles
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Game Theory |
| By Bayo Banjoko - Email Bayo Banjoko |
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| Clinical decision making's root can be traced to game theory. This article is provides an introduction to the clinical point of view of game theory.
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Clinical decision making refers to any act of diagnosis that leads to a decision regarding prognosis, treatment and referral or counselling and is as a tool in managing acute and chronic medical conditions. The formal methodology of clinical decision making has it origins in probability theory, statistical inference, economics and game theory.
Game Theory’s introduction to decision making can be traced back to the work of John von Neumann and Oskar Morgenstern in the 1930s and 1940s. The decision tree that commonly used in decision making is a special case of their game tree and their axiomatization of utilities gave rise to the methodology used to construct utility curves.
Game theory is concerned with how rational individuals make decisions when they are mutually interdependent. This definition applies mainly to one of two types of Game Theory; non-cooperative games. This is the branch of game theory that has found itself most applicable to economics. The players in a non-cooperative game are unable to enter into a binding and enforceable agreement with one another make.
The second branch of Game Theory; Cooperative Games allows players to make these agreements. It thus shows how individual make rational mutually beneficial decisions. This characteristic is called individualism. This characteristic along with rationality and mutual independence form the three pillars that underpin game theory.
In a normal form game (strategic form) there are players. These players are the individuals that make the relevant decisions. For there to be interdependence we need to have at least two players.
Strategy is a complete description of how the player will play the game. Pay-off is a player’s reward at the end of the game which is dependent on all players in the game.
This paper is centred on a particular type of two-player, non-zero sum game known as the Prisoners Dilemma. This came about as a theoretical problem concerning two prisoners who have been incarcerated for an offence that they are suspect to have committed together.
There is insufficient proof to convict them. They are isolated from each other and interrogated individually.
The consequences of there actions are explained to them.
1. If one suspect confesses and his partner does not, then the one that confesses turns state evidence and goes free and the other goes to jail for twenty years.
2. If both suspects confess they both go to jail for ten years.
3. If both suspects remain silent then they both go to jail for a lesser charge.
The dilemma lies in the fact that each prisoner has a choice between two options, but cannot make a good decision without knowing the others choice.
Tit-for-tat
Axelrod in 1980 invited game theorists to submit computer programs that best performed the task of playing a game of the Prisoners Dilemma repeatedly. The programs were then randomly paired in a tournament against each other. Each pair of contestants then repeated the game between each other two hundred times and the whole process repeated five times and the average pay-off calculated.
Of the 14 entrants the most successful strategy was Tit-for-tat submitted by Anatol Rapoport. It was deemed the simplest strategy and outperformed more complex strategies. The program had three properties that made it successful - niceness, provacability and forgiveness.
The strategy co-operated at first play then it copied its partner’s previous play. It has a short memory. It would not remember that its partner had failed to co-operate for long, but it punished its partner for not co-operating by not co-operating on the next play. This is the strategy adopted by the Prisoners Dilemma website at http://serendip.brynmawr.edu/playground/pd.html
On playing the game the game adopts the tit-for-tat strategy. The investigation involves using different strategies to play the game to determine the superior.
The game calculates our score at the end of the game. Two strategies, one of pure defection and another of pure cooperation where played against the game.
Because the computer strategy is optimal under the widest possible set of strategies it will not necessarily succeed at every game but over several iterations it will be shown to be the most successful against all other strategies.
The iterate Prisoner’s Dilemma is regarded as a standard model for the evolution of cooperation. From the literature we would expect that the computer to win as the Tit–for-Tat strategy is superior to any other.
If we were to play the computer with a Tit for-tat strategy the result would be the same as for pure co-operation as the computer iterates our last move. In this case we tie with the computer but the rewards for us both as individuals and together are the highest out of all the strategies used.
From this very limited experiment we can conclude that the Tit-for-tat strategy is superior even when faced with each other. If both players cooperate an equilibrium will be reached were both will cooperate as the punishment in the long run will be the loss of cumulative score. When all players act cooperatively they do better than when all of them act uncooperatively.
But for a fixed strategy of the player(s), a player will always do better by playing uncooperatively than playing co-operatively.
Game theory has been used to examine the relationship between health, poverty and development in developing countries, especially in how they are relate to infectious diseases such as HIV.
It has also been used in providing a representation of the interactions that take place in an operating environment in order to optimize the functioning of an operating room. It has also for use in the analysing management options in gastro-oesophageal reflux disease.
Other applications of the prisoners Dilemma can be found in Strategic Trade Policy between two countries. The Prisoners Dilemma may apply in that both countries will be worse of in the absence of cooperation.
A situation in which game theory can used, especially in the United Kingdom where waiting lists for hospital admissions are a problem, is in hospitals with an apparent bed shortage. Doctors are asked to cut down on the number of admissions. If each doctor was to consider his or her own self-interests then no beds will be saved.
REFERENCES
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2. Watt, S. Clinical decision-making in the context of chronic illness. Health Expect. 2000 Mar; 3(1):6-16.
3. Cantor, S. B. Decision analysis: theory and application to medicine. Prim Care. 1995; 22(2):261-70
4. Davies, M. D. Game Theory: A Non-technical Approach. Dover Press. 1997.
5. Shafer, Glenn. Decision Making. [WWW] http://www.glennshafer.com/assets/downloads/rur_chapter3.pdf ( 10TH Sept 2003)
6. Romp, G. Game Theory: Introduction and Application. New York : Oxford University Press. 1997.
7. Axelrod, R. Effective Choice in the Prisoner’s Dilemma. Journal of Conflict Resolution. 1980; 24:3-25.
8. Brembs, B. Chaos, cheating and cooperation: potential solutions to the prisoner’s dilemma. OIKOS. 1995; 76:14-24.
9. Folch, E.; Hernandez, I.; Barragan, M.; Franco-Paredes, C. Infectious diseases, non-zero-sum thinking, and the developing world. Am J Med Sci. 2003 Aug; 326(2):66-72.
10. Marco, A. P. Game theory in the operating room environment. Am Surg. 2001 Jan; 67(1):92-6.
11. Sonnenberg, A. Special review: game theory to analyse management options in gastro-oesophageal reflux disease. Ailment Pharmacol Ther. 2000 Nov; 14(11); 1411-7.
12. Weller, P. Lecture Notes: Advances In Systems Modelling. City University , London . 2000.
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