Thursday, 25 April 2013

                           Game Theory with Reference to Agricultural Markets
                                                  by Dr.Munish Alagh

  1. Games and Agriculture-A theoretical introduction.

1.1  Introduction

The classic review of Game Theory with Reference to Agricultural Markets by Sexton(1994) is used below to introduce the applications of Game Theory to Agricultural Markets. However other references in Game Theory including certain classical ones[1] can also be alternatively used.

The methodology of Game theory has not been much used by Agricultural economists. This is mainly because Agricultural Economics is an applied field and game theory is a tool of economic theory. Another factor may be that agricultural markets are regarded as prototype competitive markets and game theory is a tool for imperfect competition.

However theory guides application and imperfect market study be it for the concern for monopsony or oligopsony power considering that the raw product is relatively immobile and the fewness of processors, at the level of powerful retail chains market power again becomes relevant, producers in agriculture are also often encouraged to form coalitions for procuring inputs and marketing, all this makes the study of game theory for agriculture relevant.

1.2   Basic Concepts and Classifications

1.2.1        Cooperative and Noncooperative games.

Games are partitioned into two broad classes: cooperative and non-cooperative. Players in noncooperative games can make binding commitments, whereas in non-cooperative
games they cannot. This distinction must be interpreted narrowly.[2] Non-cooperative games are analysed in either their normal or extensive form. The extensive form is manifest as the familiar game tree. The normal or strategic form is a summarized description of the extensive form.

1.2.2        The Extensive Form

Figure 1 is a simple model of moral hazard. There are two players a farmer (the principal) and a marketer (the agent).If the farm product is marketed effectively ( spoilage), it is worth 3.0 at retail. A marketing agent can provide these services at a cost of 0.5, or the farmer, who is less efficient at marketing, can provide them at a cost of 1.0. The farm product net of marketing costs is worth 2.5 if the agent expends a high effort in marketing it. Assuming that there are many competitive agents,so that agents services are priced at cost. The product is worth 2.0 if the farmer vertically integrates and markets the product himself. The product is worth only 1.5 if the agent shirks and expends low effort.

The cornerstone solution concept for noncooperative games is the Nash equilibrium.Taking his opponent’s strategy as a given ,if no player would like to change his action, the resulting strategy combination is a Nash Equilibrium.

Information and Extensive Form Games
A players information set at any point in the game consists of the different nodes in the game tree that he/she knows might be the actual node but cannot distinguish among by direct observation. Consider the simple Coordination problem among farmers illustrated in the Figure. There are two market periods, early and late, and either farmer can plant a perishable crop for harvest during one but not both periods. The early harvest period is more lucrative due to greater demand, and Farmer A who runs a larger operation is better able to take advantage of the early market than is Farmer B. However, if the farmers can coordinate their plantings to smoothen supply across market periods, they will each do better than if they harvest for the same period and create a glut. A similar coordination story might involve scheduling harvests to best utilize fixed processing capacity. The payoffs under the alternative outcomes  are listed  at the end nodes in Figure 2.

Panels a) and b) in Figure 2 illustrate to alternative ways this game might be played. In panel (a) the players commit to planting decisions simultaneously. Thus, although Farmer A is depicted first on the game tree, Farmer B does not know A’s choice when it is time to make his/her own choice, ie. He/she does not know whether B1 or B2 is the actual node. His information set consists of (B1, B2). Information sets are depicted on game trees by either encircling nodes that comprise an information set as in panel(a) or connecting the nodes with a dashed line.

Panel (b) depicts a case where Farmer A is able to move first. How he/she achieves this position might be an interesting strategic question. For example he/she could sign a labour contract specifying an early planting cycle and containing a large penalty for breach. In this case Farmer B knows what action farmer A has taken when it is time to make his/her decision. Every information set in panel B consists of a single set or in game theory parlance is a singleton.

Figure 2 illustrates the distinction in game theory between perfect information and imperfect information. In a game of perfect information each information set is a singleton; otherwise it is a game of imperfect information.

What are the pure strategy Nash equilibria to the coordination games in Figure 2? The game in panel (a) has two equilibria for (A,B): (EARLY, LATE) and (LATE, EARLY). The total payoff from (EARLY,LATE), exceeds that from (LATE, EARLY), but there is no way in this non cooperative game structure for Farmer A to necessarily persuade Farmer B to undertake that option.

Farmer B’s strategy choices are complicated somewhat in the game depicted in panel b). They must specify his/her move in response to either of A’s possible actions. Three Nash equilibrium strategy combinations emerge:
1.      (EARLY, if EARLY then LATE; if LATE then EARLY) with outcome that A plays EARLY and B plays LATE.
2.      (LATE, if EARLY then EARLY; if LATE then EARLY) with outcome that A plays LATE and B plays EARLY.
3.      (EARLY, if EARLY then LATE; if LATE then LATE) with outcome that A plays EARLY and B plays LATE.

An important refinement of Nash equilibrium is the concept of subgame perfect equilibrium due to Selton (1975). The game depicted in Figure 3b) is dynamic in that A moves first and B observes his/her move. Yet the construct of Nash  equilibrium requires A to take B’s strategy as given in choosing his/her own move. This fact tends to produce Nash equilibria in dynamic games that involve noncredible threats on the part of some player(s). Both the second and third equilibrium to the game in panel (b) involves such threats. Equilibrium 2 involves a threat by B to play EARLY regardless of A’s action. Taking this strategy as given A’s best reply is LATE. However, if A chose EARLY so that it was fait accompli, B’s optimal response is to choose LATE, not EARLY. Similarly, the threat to play LATE if LATE in equilibrium 3 makes no sense, yet because B is never called upon to make that move in equilibrium, the strategy combination is a Nash Equilibrium.

Subgame perfection works to eliminate noncredible threats. To understand the concept it is necessary to define a subgame. A subgame is a game consisting of a node that is a singleton for all players, that node’s successors and the payoffs at the associated end nodes. The game in Figure 2b) has three subgames: the complete game itself and the games beginning at nodes B1 and B2. Conversely in panel A the only subgame is the game itself.

A SPE is a set of strategies for each player such that the strategies comprise a Nash equilibrium for the entire game and also for every subgame. Subgame perfection requires strategies to be in equilibrium everywhere along the game tree, not only among the equilibrium path.

The concept is exceedingly useful for analyzing dynamic games of perfect information such as those depicted in Figures 1 and 2b and also games of ‘almost perfect’ information. These are dynamic games where at given date t players choose actions simultaneously knowing all actions taken during the preceding periods. The within periods simultaneity is a deviation from perfect information. The most common example of these games are repeated games where players repeatedly play a simultaneous single period game, such as a prisoner’s dilemma or choices of price or quantity by oligopolists in a static market environment.

1.2.3. Games of Incomplete or Imperfect Information.

Let us introduce Nature as a player who moves first at the outset of a game.The choices made by Nature define a player’s type, including possibly his/her strategy set, payoff functions, and knowledge concerning locations on the game tree-information partitions in game theory parlance. When nature moves in these environments, this is said to establish a state of the world.

Figure 3 illustrates the modeling process for the sequential-choice version of the coordination game among farmers. The incomplete information concerns player B’s type. He might be either a “profit-maximiser” or “mean-spirited”. A profit-maximising B has the same payoffs as in Figure 2. A mean-spirited B, however obtains utility from inflicting pain upon his/her neighbor, and hence will always time his planting to diminish A’s payoff. The way to model this uncertainty is to let Nature choose  between (maximiser, mean) with probabilities (P, 1-P).

Moves by Nature at the outset of a game convert the game to one of incomplete information when at least one of the players is uninformed of natures choice. If some players observe nature’s choice and others do not, then the game involves assymetric information, and some players have valuable private information.

In Figure 3 the more sensible alternative is that A is uninformed which produces the extensive form in Figure 3a. The less realistic alternative in this particular example but the alternative with more important consequences for game theoretic modeling is that B is uninformed as illustrated in Figure 5b). The dotted lines depict information sets which are not singletons. In Figure 5a) Farmer A does not know Nature’s choice, and hence, whether the actual node is A1 or A2. Player B’s information sets are all singletons because he observes both Nature’s and A’s move.

The type of game depicted in Figure 3b is interesting because it possibly allows the uninformed player to update his/her information based upon the informed player’s move. This type of scenario has prompted further refinements of Nash equilibrium. Such a refinement is perfect Bayesian Equilibrium (PBE). In a PBE players strategies are optimal given their beliefs and beliefs are obtained from strategies and observed actions using Baye’s rule whenever possible.

The following is a formal definition of a PBE based on Ramunsen (1989): A PBE consists of a strategy combination and a set of beliefs such that at each node of the game:1) the strategies are Nash for the remainder of the game, given the beliefs and strategies of the other players, and 2) the beliefs at each information set are rational given the evidence, if any, from previous play in the game.

1.2.4 Further Refinement
Another equilibrium concept that was developed contemporaneously with PBE and has similar properties is Selton’s (1975) concept of trembling-hand perfect equilibrium. The idea behind trembling hand perfection is that players may make mistakes (their hands may tremble) during play of a game. A trembling hand perfect equilibrium strategy continues to be optimal for a player even if there is a small chance that some other player will pick an out-of-equilibrium action.
For an example of how trembling-hand perfection refines equilibrium consider the coordination game between farmers in figure 3b). One Nash equilibrium involves A, who moves first, playing EARLY and B playing (if EARLY than LATE; if LATE than LATE). As long as  A plays EARLY, B’s strategy is a best reply, but if there is a chance that A will tremble and play LATE, then it is certainly not optimal for B to respond with LATE, ie., this NASH equilibrium is not trembling hand perfect. The equilibrium where A plays LATE and B plays (if EARLY then EARLY, if LATE then EARLY) can be eliminated by the same argument.

2.  Game theory applications

Game-theory is relevant when markets are imperfectly competitive and Sexton (1994b) argues that this condition is commonly met in agriculture. Specific topics of application include principal-agent models, auctions and bargaining.
Seller market power may be important at most levels of the food chain, except the raw-product (farm) level. Another important dimension of imperfect competition in agricultural markets may be monopsony, oligopsony power exercised by processors and handlers over farmers. Because agricultural products are often bulky and/or perishable, they are costly to transport. Thus, markets for raw agricultural products are spatial markets, an arena where imperfect competition is almost certain.

Imperfect competition is also the norm in the international trade of many agricultural products. In large part this condition is caused by the intervention of marketing boards and state trading companies to govern export trade and centralized import authorities to control purchases of food products. An extensive game-theory based strategic trade literature has arisen to analyse imperfect competition in trade. (See, Krishna and Thursby, 1990, for an important survey.)
Imperfect information and uncertainty also represent important departures from perfect competition in agricultural markets. Uncertainty opens the door to strategic markets particularly when the uncertainty or lack of information is assymetric across agents. Such informational assymetries may be significant in agricultural markets. For example, processors are probably often better informed about market demand conditions than are farmers. Processors may have incentives to exploit these informational advantages, whereas farmers have incentives to encourage processors to reveal truthfully their knowledge of market conditions.

By the same token, farmers in many cases will have informational advantages over processor handlers concerning their charcteristics as growers. In the simplest signaling model context, a grower might be LOW or HIGH quality, with HIGH-quality growers problem being to signal their type to processors, while LOW-quality types try to masquerade. Characteristics of the agricultural product itself are an issue in many contexts, opening the door to interesting adverse selection problems. Although product characteristics are important, they become a subject for game theory only when information as to characteristics is assymetric, e.g, the handler knows whether the produce is fresh, but the retailer does not and verification is costly.

Applications discussed in this review are what may be called vertical exchange mechanisms. They are: Principal-agent models with assymetric information, auctions and collective bargaining.

2.1 Principal-agent models
The principal is the entity who hires the agent to perform some task. In almost all cases, the agent acquires an informational advantage at some point in the game as to his/her type, actions, or other states of the world. Contexts for applications of this basic model in agricultural markets may be several. Some applications may involve the farmer or grower seeking to contract with a marketing firm as agent to sell his/her production. The agent may have specialized knowledge as to his/her own ability, market conditions. Alternately, a processor/handler may be modeled as the principal who seeks farmers to grow products to his/her specifications. Growers may have specialized knowledge as to their types, production costs etc.

Potential applications of the model need not be limited to the first-handler level either. It may be useful, for example, to model the behavior of a large retail food chain seeking manufacturers of private-label products as a principal and the manufacturer as an agent. Or in some contexts it may be useful to consider a manufacturer as the principal and retailing firms as the agents.

The models can be partitioned according to the nature of information asymmetry. Models where the agent takes actions unobserved by the principal are known as moral hazard problems. Models where the agent has hidden knowledge prior to contracting with the principal are known as adverse selection models. Adverse selection models nay involve signaling, with the agent taking actions to signal his type from the  principal.

2.1.1 Moral Hazard Models. We frame the moral hazard problem in the context of a grower seeking a marketing agent to handle his/her production. In most principal-agent problems with moral-hazard the unobserved action is referred to as the agent’s effort. The term must be interpreted broadly. In the context of a marketing firm, effort could refer to speed of transit to market for sake of freshness, proper refrigeration to retard spoilage, advertising and promotion activities, diligence in processing etc.

The essence of the moral hazard problem is that if given the opportunity, the agent accepted a contract and expended low effort, causing the grower to elect the grower to elect to market the product himself at a cost in terms of inefficiency. The problem arose because the grower could not observe the agent’s level of effort (ie the action was hidden). A more sophisticated version of the moral hazard model is obtained by assuming that, though effort is unobservable, a variable related to effort is observable. This variable may be profits, the level of output, or the per-unit price that the grower receives net of any marketing costs.
In this case the problem is to design a contract based on the observed variable to elicit the optimal expenditure of the unobserved variable-effort.
For reputation to have its effect, the model must be specified with incomplete information. For example, if the principal retains even a slight probability that the agent is predisposed to produce high quality or effort, the agent has incentive to actually produce high quality or effort to perpetuate that perception at least until the latter plays of the game.
This framework may yield valuable insights regarding contract structure in agriculture when the processor/handler is modeled as the principal and the grower as the agent. For example, product quality dimensions are increasingly important in today’s food market. Raw product quality can be influenced by farmers horticultural practices (effort), but it is also influenced by random factors that cannot be observed perfectly by the processor. Depending upon the raw product and the nature of the harvest technology, aspects of product quality may be discerned directly through grading. The processors job in these cases is to specify contracts with growers that solicit the processor’s desired quality level subject to incentive compatibility with growers and also their financial viability. Imperfect monitoring may involve inability to observe directly either farmers horticultural practices or the characteristics of the harvested product.
Contractual practices may vary widely across raw agricultural product markets, and much of the variation in contracts  may deal with differences across markets in the importance of and the variability in product characteristics and, in turn, on the extent to which these characteristics can be monitored by observation of the product or grower’s horticultural practices.
2.1.2 Adverse Selection- If the marketing sector is at various stages is unable to recognize and reward quality, the message of the adverse selection models is that high quality will be driven out.The pooling practices of cooperatives are especially worrisome in this regard. If cooperatives are less able to reward quality than other organizational forms, the equilibrium configuration across organisations calls for predominantly low-quality producers to patronize cooperatives.
In agriculture, the various quality provisions mandated by marketing orders and marketing boards may be justified as a response to adverse selection. If not for adverse selection, quality standards that proscribe products with certain characteristics merely limit consumer choices. With assymetric information, however, failure to impose quality standards also limits choice by driving out high quality.
2.1.3 Vertical control problems-For instance, Landlord tenant and Processor retailer transactions regarding the manner of control by the principal(Landlord/Processor(food manufacturer in dealing with Tenant/retailer(the agent).
But with large retailers emerging the role may be reversed , a very important instance in this context is the case of slotting allowances charged by retailers to carry the manufacturers product.If quality information is assymetric with manufacturers having information and retailers not having it, then manufacturers with low quality may be tempted to not pay slotting allowances, also high quality manufacturers will be willing to pay slotting allowances for test markets as well.
2.1.4 Auctions workin agriculture under condition of volatile prices like fresh fish, eggs and some fruits and vegetables where posted prices work poorly  also in cases of variable quality like livestock, wool where also posted prices work poorly, often spatial factors make electronic auctions relevant, in non competitive cases the example is of Government auctioning the right to obtain subsidy.
2.1.5 Collective Bargaining can take place by government fiat or voluntary initiative of growers. Bargaining is an important application of non cooperative game theory. Information is an important aspect in Bargaining Models.

3.1 Strengths and Weaknesses of Game Theory: The Case of Land Rental Contracts
We focus on a particular problem to illustrate some of the strengths and weaknesses of game theory for agricultural economic problems. For this purpose, we focus on the farm rental contract between a landlord and tenant. Contract choice is particularly appropriate as a game, but its game-theoretic aspects are rarely recognized. Suppose the landowner can offer one of two contracts. In contract A, the tenant gives the owner two-thirds of the crop. In contract B, the tenant gives the owner one-third of the crop and a fixed payment, K. The tenant can take one of three actions: sign the contract and use a high level of inputs, sign and use a low level of inputs, or not sign the contract. Crop yields are high and low if the contract is signed, respectively. If the tenant does not sign the contract, the owner gets zero payoff. Output price is normalized to 1 and the input price is given. With these specifications, the game has four equilibria: 1. {low input, if A, high input, if B; contract B } 2. {low input if A, low input if B; contract A } 3. {low input, if A, no contract if B; contract A} 4. {no contract if A, high input, if B; contract B }where the first component is the tenant's strat-egy and the second is the landlord's strategy. Each equilibrium specifies a particular choice of strategy for both players. For example, in equilibrium 1, the tenant's strategy is to choose input level low if the landlord offers contract A and input level high if the landlord offers contract B; the landlord's strategy is to offer contract B. Equilibrium 1 (and its analog in a variety of games) is the one on which the literature almost invariably focuses, and the predicted contract is B. Agricultural economists have long observed the superior qualities of contract B. Allowing the tenant to keep a higher proportion of returns leads to higher input use, higher yields. Con-tract B occurs in equilibria 1 and 4. Although not often recognized, this game has two equilibria in which contract A is chosen. Some equilibria can be ignored by implicitly or explicitly invoking two restrictions: perfection and properness. Perfection requires that the proposed equilibrium be an equilibrium of all subgames. There are two proper subgames of the figure 1 game in which only the tenant moves. Equilibrium 2 is not perfect because the tenant "threatens" to play low input if B, yet the tenant is better off with other actions if B were actu-ally offered. The desire to rule out such noncredible threats led Selten to formalize per-fection. Perfection also rules out equilibrium 4. The perfect equilibria are 1 and 3. The most unusual equilibrium is 3. It arises because contract B leaves the tenant indifferent between accepting and not accepting the con-tract, but if he rejects the contract, the owner should choose contract A. In summary, this game has two perfect, proper equilibria yield-ing contract A or B, and no standard procedure is available for predicting one over the other.

[1] Luce and Raiffa(1957), Kreps.D (1990a, 1990b) and Watson (2004) are some texts this author has profitably used.
[2] For details see Sexton (1994).

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