game theory applied to a policy problem of rice farmers

8/12/2019 Game Theory Applied to a Policy Problem of Rice Farmers

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Agricultural Applied Economics Association

Game Theory Applied to a Policy Problem of Rice FarmersAuthor(s): Max R. LanghamSource: Journal of Farm Economics, Vol. 45, No. 1 (Feb., 1963), pp. 151-162Published by: Oxford University Presson behalf of the Agricultural & Applied Economics

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152 MAX R. LANGHAMthe government s prohibited romadjusting upportpricesto levels lowerthan 70 percent of parity. They can, of course, adjust parity price up-ward, but there are risks of surplus accumulations if higher prices resultin increased production. With these considerations, it is assumed that thegovernment would not increase parity prices above 85 percent.The payoff matrix

Published input-output data [7] were used to compute coefficients forthe payoff matrix. In the decision model the typical rice farm (selectedto represent Louisiana rice farmers) has 400 acres, 120 of which are al-lotted to rice. An average yield of 20 barrels of rice per acre is expectedand is to be grown under the typical rice-native grass system of rotation.

TABLE 1. PAYOFF MATRIX FOR RICE FARMERS' POLICY PROBLEMaRice farmers' Government'salternatives(% acreage% acreage (percentof parity)allotmentincreaseo 70 73 76 79 82 85 Row minimaask for)

0 -1,982.40 -1,284.00 -583.20 117.60 816.80 1,516.80 -1,982.4011 -1,833.10 -1,123.06 -410.58 301.90 1,011.94 1,724.34 -1,833.1031 -1,683.80 - 962.12 -237.96 486.20 1,207.88 1,932.04 -1,683.805 --1,534.50 - 801.18 - 65.34 670.50 1,403.82 2,139.66 -1,534.5061 -1,385.20 - 640.24 107.28 854.76 1,599.76 2,347.28 -1,385.2081 -1,235.90 - 479.30 279.90 1,039.10 1,795.70 2,554.90 -1,235.9010 -1,086.66 - 318.36 452.52 1,283.40 1,991.64 2,762.52 -1,086.66112 - 937.30 - 157.42 625.14 1,407.70 2,187.58 2,970.14 - 937.30131 - 788.00 3.52 797.76 1,592.00 2,383.52 3,177.76 - 788.0015 - 638.70 164.46 970.38 1,776.30 2,579.46 3,385.38 - 638.70Columnmaxima - 638.70 164.46 970.38 1,776.30 2,579.46 3,385.38

a Governmentalternatives71, 72, 74, 75, 77, 78, 80, 81, 83 and 84 are omitted from this tablefor simplificationof the presentation.It is assumed that existing resources are sufficient to permit up to a 15 per-cent rice acreage increase without further fixed investments or withoutserious sacrifice to other existing enterprises.3 Only operating costs andrepairs are assumed to be incurred.4The payoff matrix for the typical rice farm is given in Table 1. Payoffcoefficients represent changes in a typical rice farmer's net income asso-ciated with the various alternatives of the players.If the game theory criteria mentioned above are applied to this payoffmatrix, the solution is trivial. This triviality stems from the fact that eachrow and column of the matrix is linear with respect to both the govern-ment's and farmers' alternatives, respectively. Consequently, column 70

3In the rice area of Louisiana most of the land not seeded to rice is permitted togrow up in native grasses. These grasses are partially grazed by beef cattle; however,it is doubtful if a 15 percent rice acreage increase would lead to reduced beef pro-duction.Additional fixed cost would be incurred if the acreage increase persisted over along period of time. Therefore, the model described here is potentially capable of an-alyzing only short-run implications of a rice acreage allotment increase.

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GAME THEORYAPPLIEDTO RICE FARMERS 153dominates every other column and row 15 dominates every other row, andthe payoff coefficient at the intersection of 70 and 15 represents a saddlepoint, i.e., a stable solution. A column is said to dominate another columnif the coefficient in each of its cells is smaller than the corresponding co-efficient in the dominated column. Similarly, a row is said to dominateanother row if the coefficient in each of its cells is larger than the cor-responding coefficient in the dominated row. These definitions are in-tuitively obvious since the government or the farmers would never (as-suming they want to maximize profits or minimize costs) choose an al-ternative in which they would always be worse off regardless of thechoice of the other.

Since each of the player's alternatives is linear with respect to his op-ponent's, only the corner cells of the matrix can be of relevance for de-cision making. The payoff matrix, showing only the four relevant cornercells, is as follows:Rice farmers'ice farmers' Government's alternativesalternatives (%acreage allotment percent of party) Row minimaincrease toask for) 70 85

0 -1982.40 1516.80 -1982.4015 - 638.70* 3385.38 - 638.70

Column maxima - 638.70 3385.38

Ignoring what is known about dominance, the four corner cells are ab-stracted so that the four decision criteria may be presented for illustrativepurposes.Wald criterion

According to Wald's criterion, the farmers should select the maximumof the row minima. The government, in turn, should select the minimumof the column maxima. In this case, a saddle point represented by thestarred number at the intersection of 15, 70 insures a stable solution.Savage criterion

In trying to minimize their maximum regret with the Savage criterion,farmers should, as with the Wald criterion, ask for a 15 percent acreageincrease. The relevant regret matrix is:Hurwicz criterion

Hurwicz proposed that the decision-maker consider a weighted average



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MAX R. LANGHAMRice farmers' Government's alternativesalternatives (% (percent of parity) Row maximaacreage allotmentincrease to ask for) 70 85

0 1343.70 1868.58 1868.5815 0 0 0*

of the best as well as the worst possible outcomes. With a as the weightattached to the row minima, this criterion would select the larger of eithera(-1982.40) + (1 - a) 1516.80, or a(-638.70) + (1 - a) 3385.38. Again,the farmers should ask for a 15 percent acreage increase regardless of thea chosen as long as 0


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156 MAXR. LANGHAMTABLE 2. HYPOTHETICAL PROBABILITY MATRIX USED TO ADJUST THE PAYOFF MATRIX OFTABLE 1 TO OBTAIN AN ADJUSTED PAYOFF MATRIXaRicefarmers' Government's alternativesalternatives (percent of parity)(% acreage Totalallotmentincrease to 70 73 76 79 82 85ask for)

0

10

13413S?

.000977

.001953

.003906

.007812

.015625

.031250

.062500

.125000

.375008

.007812

.015625

.031250

.062500

.125000

.265686

.281281

.125000

.062500

.062500

.125000

.252442

.254151

.125000

.062500

.031250

.015625

.007812

.252442

.125000

.062500

.031250

.015625

.007812

.003906

.001953

.000977

.031250

.015625

.007812

.003906

.001953

.000977

.000488

.000244

.000122

.003906

.001953

.000977

.000488

.000244

.000122

.000061

.000031

.000015

1.0000001.0000001.000000

1.0000001.0000000.999999

1.0000001.0000011.000000

15 .750008 .031250 .003906 .000488 .000061 .000008 1.000000a Probabilities associated with government alternatives 71, 72, 74, 75, 77, 78, 80, 81, 83and 84 are omitted from this table for simplification of the presentation.

Adjusted Payoff MatrixLet the payoff matrix in Table 1 be represented by [aij]m,n and theprobability matrix in Table 2 by [pij]m,n, then the adjusted payoff matrixin Table 3 is given by [aijpij]m,n.That is, each cell in Table 1 is weightedby the probability in the corresponding cell of Table 2 to give Table 3.Viewing Table 3 as a realistic representation of farmers'payoff expecta-tions, the decision problem may again be appraised in what we considera more realistic manner.

Wald criterionThe maximum of the row minima of the adjusted payoff matrix would



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GAME THEORYAPPLIEDTO RICE FARMERS 157

TABLE 3. ADJUSTEDPAYOFFMATRIXaRice farmers' Government's alternatives (percent of parity) Row sum Decisionalternatives Row indexes(% acreage Row n 1allotment minima 2 ai 2 aincrease to 70 73 76 79 82 851 16ask for)

0 - 1.9 - 0.03 -36.45 29.69 25.50 5.92 - 43.80 - 5.36 - 0.341 - 3.58 - 17.55 -51.32 37.74 15.81 3.37 - 51.32 - 55.00 - 3.44Si - 6.58 - 30.07 -60.07 30.39 9.44 1.89 - 60.07 -111.98 - 7.005 - 11.99 - 50.07 -16.61 20.95 5.48 1.04 - 78.73 -174.76 -10.926{ - 21.64 - 80.03 13.41 13.36 3.12 0.57 -100.43 -241.27 -15.088? - 38.62 -127. 4 17.49 8.12 1.75 0.31 -127.34 -308.91 -19.3110 - 67.92 - 89.55 14.14 5.01 0.97 0.17 -161.58 -373.70 -23.3611 -117.16 - 19.68 9.77 2.75 0.53 0.09 -211.68 -433.41 -27.0913? -295.51 0.22 6.23 1.56 0.29 0.05 -295.51 -491.06 -30.6915 -479.03 5.14 3.79 0.87 0.16 0.03 -479.03 -504.76 -31.55Columnmaxima - 1.94 5.14 17.49 37.74 25.50 5.92Aa Government alternatives 71, 72, 74, 75, 77, 78, 80, 81, 83, and 84 are omitted from this table for simplificationof the presentation.

require that farmers ask for no acreage increase. This value does not,however, correspond to the minimum of the column maxima, and thesolution is not a stable one. However, with a mixed strategy, a stable solu-tion with the Wald criterion does exist.With a mixed strategy, farmers should choose their various alternativeswith specified probabilities. In this case, the farmers' best mixed strategyas computed5 from the payoff matrix indicates that farmers should playtheir alternatives with the following probabilities:6Farmers should ask for an

acreage allotment increase of01%33'56%8%1011%

13315Total

With a probability of.6843.0000.0339.0747.0402.0704.0072.0650.0000.0239.99967

' For computationalproceduressee reference [1, pp. 512-27].'The randomizationof a strategy is a relatively simple matter for the players.One method would be to mark off a disk into pie-shaped pieces whose arcs areproportional to the given probabilities. A spinner at the center of the disk couldthen be used to determineplayers'moves.7Since the probabilitiesdo not sum to one, a slight rounding erroris indicated.



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MAX R. LANGHAMIn response the government should randomize its strategy with thefollowing probabilities:

Government should set thesupport price at the fol-lowing percent of parity

707172737475767778

855

With a probability of.0534.0640.0730.1082.1104.1863.0672.3373.0000

.0000Total .9998

If the farmers follow the conservative mixed strategy given by theWald criterion and the government responds with its best strategy, thetypical farmer could expect to lose $25.62 (the value of the game). Andfarmers would be expected to ask for a weighted (by the probabilities ofthe mixed strategy) average acreage increase of 2.5 percent. The weightedaverage support price that the government would be expected to respondwith is 74.7 percent of parity. The typical farmer could expect to lose$18.18 less ($43.80 - $25.62) by choosing a mixed rather than a purestrategy.Savagecriterion

An adjusted regret matrix (omitted in this paper) can be derived fromthe adjusted payoff matrix. Each element in such a regret matrix wouldrepresent the amount that must be added to the corresponding element inTable 3 to equal the maximum element in the same column of Table 3.Choosing the pure strategy that minimizes the maximum regret,farmers should request no acreage increase. Again this pure strategy doesnot present a stable solution. The simplex solution for a mixed strategyassigns the following probabilities to the farmers' alternatives:

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GAME THEORYAPPLIE TO RICE FARMERSFarmers should ask for an

acreage allotment increase of01%3%56%8%1011%13%315

With a probability of.4864.0000.0846.0753.1175.0031.1074.0920.0000.0338

Total 1.0001With this randomized strategy, a typical rice farmer would have aminimized regret with a weighted average expected acreage increase of4.1 percent. He would be expected to receive a minimized maximum re-gret of 62.62,3.57 less than with a pure strategy.The 4.1 percent weighted average increase that farmers would ask forusing the Savage criterion is less conservative than the 2.5 percent withthe Wald criterion.

HurwiczcriterionThe application of the Hurwicz criterion to the adjusted payoff matrixwould lead rice farmers to that strategy representing the maximum oftheir pessimism-optimism indexes:

Pessimism-optimism indexesa(- 43.80)+(1-a)(- 51.32)+ (1-a)a(- 60.07)+ (1-a)a(- 78.73) + (1-)C(-100.43) + (1-a)a(-127.34) + (1-a)a(-161.58) + (1--a)a)-211.68) + (1-a)a(-295.51) + (1-a)a(-479.03) + (1- a)

43.80 = 43.80 - 87.60a37.74 = 37.74 - 89.06a30.40 = 30.40 - 90.47a26.47 = 26.47 - 105.20a22.22 = 22.22 - 122.65a17.49 = 17.49 - 144.83a14.14 = 14.14 - 175.72a11.41 = 11.41 - 223.09a8.44 = 8.44- 303.95a6.80 = 6.80 - 485.83aA graph of these indexes (Figure 1) shows that any a, 0< c


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MAXR. LANGHAM

x 30 \

1

; 15\ 8.1 .2 .3 .4 .5 1.0

aFIGURE 1. PESSIMISM-OPTIMISM INDEXES AS A FUNCTION OF ( FOR THE

DECISION PROBLEM OF RICE FARMERS.It is conceivable that in a policy problem a could be considered as afunction of the decision-maker's alternatives; i.e., a would be formulatedas a variable which takes on different values for the various alternativesof the decision-maker. Such a formulation would seem especially appro-priate if one chose to apply the Hurwicz criterion to the original payoffmatrix in Table 1. For example, if in the original matrix of Table 1 thedecision-maker had considered an a of 4 appropriate for the no acreageincrease alternative and an a of 1 appropriate for the alternative of a 15percent acreage increase, the original decision would have been reversed.And, the choice of a different a for the two alternatives leads to a decisionmodel that more realistically reflects the more commonly held notion ofthe farmers' decision processes.Laplacecriterion

With this criterion, the farmers would ask for no acreage increase since-.34 is the largest decision index (Table 3).A modifiedBayessolution8The Laplace criterion may be viewed as a special case of the Bayessolution. The Laplace criterion assumes there is insufficient reason or

8 [6, p. 16]; [4, p. 41].One reviewer indicated that it was simply not fair to assume a hostile govern-ment since Table 2 takes care of any assumptionsabout this player. If one interprets

160



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GAME THEORYAPPLIEDTO RICE FARMERSknowledgeto associatevaryingprobabilities,pj, with the variouscolumns(alternativesof the opponent).So, the decision-maker ssumes that hisopponent plays each alternativewith equal probability.The Bayes solu-tion assumesthatknowledge s availableto permitthe associationof someprobability,pj,

(with pj > 0 and pj =with each columnof the payoffmatrix.Then, the decision-makerhoosesthat strategywhichmaximizes

nE aijpj.j=lA modifiedBayes solutionis permittedby assumingthat knowledgeisavailableto permit assigningsome probability,pij, to each element,aij,in the payoffmatrix.These probabilityelementshave the followingpro-perties:

npij >, pi = 1.j=l

They are not homogenousover i as assumedby a straightBayessolution.An applicationof the modifiedBayes solution to the adjusted payoffmatrixindicates that farmersshould select that strategywith the largestvalued index. This index is found by summingacrossthe columnsof theadjusted payoff matrix and results in farmerschoosing no acreage in-crease for their optimumstrategy(Table 3).SummaryThe Wald, Savage,Hurwiczand Laplacedecision-makingriteriawere

applied to a game theoric model in an analysis to determine whetherrice farmers should request an acreage allotment increase. A trivialpayoffmatrix,derivedfromexistingdata, and the four criteriawere usedto obtain the farmers'best strategy.With each criterion, he farmers'beststrategywas to ask fora 15percentacreage ncrease.A probabilitymatrixwas then hypothesized n orderto adjustthe orig-inal payoff matrixto obtain what was believed to be a more politicallyrealistic estimate of the relevant payoff matrix. Four decision-maKingcriteriawere then appliedto this adjustedpayoffmatrix.The conclusionswere no longerconsistent.The Wald and Savagecriteriahad stable solu-the adjusted payoff matrix in this way, the best decision criterion would be themodified Bayes solution presented here or the stable solution given by the Hurwiczcriterion-both of which give the same short-runconclusion. However, one may in-terpret Table 3 as a method of adjustingfor public opinion (a thirdplayer in theshort run consideredhere). This adjustmentfrees both players from outside pressuresand leaves them to seek their own goals. This latter interpretationgives relevance tothe mixed strategiesof the Wald and Savage criteria.

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game theory applied to a policy problem of rice farmers

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