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Distributing medical resources

An application of cooperative bargaining theory to an allocation problem in medicine

Series:

Antje Köckeritz

Allocating scarce medical resources has become an important topic in public discussion. In the German statutory health system we are facing a situation of lacking adequate funds for all needs. The financial restrictions force us to use resources wisely. This emphasizes the need of general allocation rules and criteria applied in medical allocation situations. The purpose of this work is to implement and interpret properties of cooperative bargaining theory to special allocation situations in medicine. The author shows how the concepts of Nash and Kalai/Smorodinsky can be applied to a medical allocation problem and discusses implications of their properties and solutions for the German health system.

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FIGURE 1.1: GERMAN STATUTORY HEALTH SYSTEM – AN OVERVIEW ............... 24 FIGURE 2.1: A BARGAINING SITUATION ..................................................................... 47 FIGURE 2.2: PROPERTY OF INVARIANCE OF POSITIVE AFFINE TRANSFORMATION .................................................................................. 47 FIGURE 2.3: PROPERTY OF WEAK PARETO OPTIMALITY ........................................ 47 FIGURE 2.4: PROPERTY OF SYMMETRY ....................................................................... 47 FIGURE 2.5: PROPERTY OF INDEPENDENCE OF IRRELEVANT ALTERNATIVES 48 FIGURE 2.6: THE NASH SOLUTION N(S,0) .................................................................... 48 FIGURE 2.7: PROPERTY OF RESTRICTED MONOTONICITY ...................................... 50 FIGURE 2.8: PROPERTY OF INDIVIDUAL MONOTONICITY ....................................... 50 FIGURE 2.9: THE KALAI/SMORODINSKY BARGAINING SOLUTION ........................ 51 FIGURE 3.1: FEASIBLE DETERMINISTIC ALLOCATIONS FOR AN ALLOCATION PROBLEM WITH 9 ,3 ,2 ,10 21 qqnq .................................... 59 FIGURE 3.2: A SET OF FEASIBLE CHANCE ALLOCATIONS ....................................... 60 FIGURE 3.3: A SET OF FEASIBLE CHANCES OF SUCCESS ......................................... 64 FIGURE 3.4: WEAK (=STRONG) PARETO EFFICIENT SET AND SYMMETRIC ALTERNATIVES ........................................................................................ 67 FIGURE 3.5: BARGAINING PROBLEM ',' rL FOR N = 2. ........................................... 69 FIGURE 4.1: FEASIBLE DETERMINISTIC ALLOCATIONS FOR EXAMPLE 1 ............ 78 FIGURE 4.2: FEASIBLE DETERMINISTIC ALLOCATIONS FOR EXAMPLE 2 ............ 79 FIGURE 4.3: FEASIBLE DETERMINISTIC ALLOCATIONS FOR EXAMPLE 3 ............ 79 FIGURE 4.4: A SET OF EXPECTED NUMBER PAIRS OF PATIENTS UNDER EXAMPLE 1 ................................................................................................. 80 FIGURE 4.5: DIFFERENT TREATMENT SETS ................................................................ 81 FIGURE 4.6: CHANGING THE REPRESENTATION FROM ALLOCATING TO PATIENTS (A) VERSUS ALLOCATING BUDGETARY UNITS (B) ......... 84 FIGURE 4.7: INDEPENDENCE OF THE BARGAINING SOLUTION OF IRRELEVANT TREATMENT PAIRS .......................................................... 85 FIGURE 4.8: THE NASH SOLUTION YIELDS IDENTICAL SOLUTIONS FOR THREE TREATMENT SETS ........................................................................ 87 12 FIGURE 4.9: TWO PROBLEMS WHEN THE NASH CONCEPT YIELDS DIFFERENT SOLUTIONS ................................................................................................. 88 FIGURE 4.10: PATIENT MONOTONICITY FOR TWO BARGAINING PROBLEMS T AND T’ ......................................................................................................... 89 FIGURE...

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