Uptake of Diagnostic Tests by Livestock Farmers: A Stochastic Game Theory Approach

Game theory examines strategic decision-making in situations of conflict, cooperation, and coordination. It has become an established tool in economics, psychology and political science, and more recently has been applied to disease control. Used to examine vaccination uptake in human medicine, game theory shows that when vaccination is voluntary some individuals will choose to “free-ride” on the protection provided by others, resulting in insufficient coverage for control of a vaccine-preventable disease. Here, we use game theory to examine farmer uptake of a new diagnostic ELISA test for sheep scab—a highly infectious disease with an estimated cost exceeding £8M per year to the UK industry. The stochastic game models decisions made by neighboring farmers when deciding whether to adopt the newly available test, which can detect subclinical infestation. A key element of the stochastic game framework is that it allows multiple states. Depending on infestation status and test adoption decisions in the previous year, a farm may be at high, medium or low risk of infestation this year—a status which influences the decision the farmer makes and the farmer payoffs. Ultimately, each farmer's decision depends on the costs of using the diagnostic test vs. the benefits of enhanced disease control, which may only accrue in the longer term. The extent to which a farmer values short-term over long-term benefits reflects external factors such as inflation or individual characteristics such as patience. Our results show that when using realistic parameters and with a test cost around 50% more than the current clinical diagnosis, the test will be adopted in the high-risk state, but not in the low-risk state. For the medium risk state, test adoption will depend on whether the farmer takes a long-term or short-term view. We show that these outcomes are relatively robust to change in test costs and, moreover, that whilst the farmers adopting the test would not expect to see large gains in profitability, substantial reduction in sheep scab (and associated welfare implications) could be achieved in a cost-neutral way to the industry.

-' 2 -(1 − -) (1 − -) ' . and the element /0 gives the probability of transitioning from state i to state j. We assume that if at least one of the farms is infested ("Both infested" or "One infested) the probability of either farm being infested next time is qH. If neither farm is infested the probability of being infested next time is qL.
The Markov chain was used to simulate an 11 year infestation history (to match the observed data) 10000 times. At the end of the 11 year infestation history, a state of "Both infested" meant that we recorded an outbreak; "Neither infested" meant we recorded no outbreak; and "One infested" meant we recorded an outbreak with a 0.5 probability. To correspond with the available data, for cases where an outbreak was recorded, we also recorded whether 1-5 or > 5 outbreaks had occurred in the previous 10 years; and for cases where no outbreak was recorded, we also recorded whether some or no outbreaks had occurred in the previous 10 years. This allowed us to obtain the probability of the observed data for any pair qH and qL and identify the maximum likelihood estimates for these parameters.

Background
The performance of a diagnostic test is defined by two concepts, i.e. test sensitivity (Se) and test specificity (Sp). Sensitivity is the probability of correctly diagnosing infestation if it is present, whereas specificity is 1-the probability of incorrectly diagnosing infestation when it is absent. Here, we can define group level sensitivity and specificity for the ELISA blood test, applied to n number of animals. The probability of detection in a flock is then The probability of obtaining a false positive in the group / flock is For the imperfect test, we assume test sensitivity Se=0.98 and test specificity Sp=0.97 (11).

Payoffs.
First consider the payoffs a farmer receives under 4 scenarios (with the red font indicating where these expressions differ from those in the main text shown for a perfect test): ① The farmer's flock is uninfested (although he doesn't know this) and he does not adopt the new blood test § P1 = RH ② The farmer's flock is infested (although he doesn't know this) and he does not adopt the new blood test The farmer's flock is uninfested (although he doesn't know this) and he does adopt the new blood test § P3 = RH -CTest -PFalse.Positive CTreat ④ The farmer's flock is infested (although he doesn't know this) and he does adopt the new blood test

Outcome probabilities.
To decide whether the farmers will consider themselves in a high, medium, or low risk state the following year using an imperfect test setting, we need to consider what the possible outcomes are this season, dependent on their actions. The four possible outcomes for a given flock at the end of the season are 1) clinical infestation is observed and treated 2) subclinical infestation is correctly identified (although the farmer does not know this) and treated 3) subclinical infestation is incorrectly identified (although the farmer does not know this) and treated 4) no infestation is correctly identified (although the farmer does not know this) and treatment is not administered.
Outcome 1 (clinical infestation is observed and treated) would happen if a flock was infested and progressed to clinical infestation but the farmer had not tested the flock, or tested and received false negatives. Outcome 2 (subclinical infestation is correctly identified and treated) would happen if the flock was infested and tested positive. Outcome 3 (subclinical infestation is incorrectly identified) would happen if the flock was uninfested and testing returns false positives. Outcome 4 (no infestation is diagnosed) would happen if the flock is uninfested and testing gives no false positives.
The vector of outcome probabilities (see Supplementary Table 2) for adopting the test A(q) and not adopting the test DA(q) using an imperfect test are (1-q)

Supplementary Table 2.
Outcome probabilities expressed in terms of the epidemiological parameters in Table 2 of main manuscript.

Transition probabilities and final pay-off matrices
Here, we present adjusted transition probabilities to account for the transitions the farmer believes will occur. For example, if the test is used and a false positive is obtained (outcome 3) on either farm, each farmer will believe his farm to be in the medium risk state the following year, whereas the actual transition would be to the low risk state if the true state of the farms is uninfested.
Transition probabilities associated with the farmer beliefs of moving between high, low, and medium risk states are defined as follows: Analogous to the perfect test case H, M and L are equal 4x4 matrices, which satisfy Lij + Mij + Hij =1, and the four possible belief outcomes for a given flock at the end of the season are (i) clinical infestation is observed and treated, (ii), subclinical infestation is correctly identified and treated, (iii) subclinical infestation is incorrectly identified and treated, and (iv) the absence of infestation is correctly identified.
If either farm experiences clinical infestation, both farms transition to the high-risk state next year (the column values refer to the outcomes (1-4) for farm 1 and the row values refer to the outcomes (1-4) for farm 2), i.e.

In the transition matrices below, the red entries indicate where the transitions based on farmer beliefs differ from the actual transitions.
If either farm believes they have seen subclinical infestation, but no clinical infestation, they believe they transition to the medium risk state, i.e.
If both farms believe no infestation was present, they believe they will transition to the low risk state, i.e.
The final pay-off matrices remain the same as specified in section 4.3 of the main manuscript.

Nash equilibrium & social optimum
In contrast to the multistate setup for the perfect test described in the main manuscript, for the imperfect test there are clear distinctions between the payoffs and transition probabilities used by the farmer for calculating the Nash equilibrium and the social optimum.
We consider two possible imperfect test scenarios: (i) Farmers are aware of the adjusted payoffs due to imperfect test and (ii) Farmers believe the test to be perfect and are not aware of adjusted payoffs. In both cases, farmers are assumed to operate on the basis of their believed transitions between the risk states. The social optimum was calculated using the adjusted payoffs together with the true transitions between states.
For the case of the farmers valuing long-term gains (discount rate, b = 1), we compared the outcome (Nash equilibrium and social optimum) of the perfect test with those of the imperfect test scenarios.
In both cases, we found that the annual incidence at the Nash equilibrium and social optimum was slightly higher for the imperfect test than for the perfect test ( Fig. S1 and Fig. S2). We also observed that when the farmer was aware of the adjusted payoffs (case (i)), the Nash equilibrium differed from the social optimum (Fig. S1, left hand side), but when the farmer is unaware of the adjusted payoffs, the Nash equilibrium and social optimum coincide (Fig. S1, right hand side). Thus, when farmers are not aware of the imperfect test sensitivity and specificity, a slightly improved outcome is obtained.