application of the game theory in function of ...

APPLICATION OF THE GAME THEORY IN FUNCTION OF DIPLOMATIC NEGOTIATING MODEL Dr.sc. Stevo Jaćimovski, Dr.sc. Dane Subošić, Dr.sc. Slobodan Miladinović, Academy of Criminalistic and Police Studies, Belgrade Abstract That the conflict would not escalate into a crisis, it must be resolved as soon as it is noticed. In addition, its solution must be optimal. One way to optimize decisions in terms of social crises and conflicts is the game theory. Game theory is a mathematical theory of conflict and crises situations. In addition, conflict and crises situations are characterized by two (or more) opposing sides, with antagonistic goals (mutually opposed and irreconcilable), where the result of each action of participant depends on what action the opponent will choose. Due to the antagonism of parties objectives involved in the negotiations, negotiation is particularly suitable for modeling activities by means of the theory of games. Of special interest to the diplomatic service is the diplomatic negotiation. In this context, this paper focuses on the modeling of diplomatic negotiation through the game theory. There are numerous strategies of conflict and crisis solving. Those strategies are: ignoring, withdrawal, domination, smoothing, compromise and confrontation. If a crisis occurs, it can be resolved (the fight until victory), solve (a compromise), reset (ignoring) or remove (change in the nature or circumstances of the entity). Model "fight to win" means defeat rival parties and can not be a good basis for establishing relations of cooperation as a new quality of relations. "Ignoring" means any failure to take steps to remedy the problem, but it is based on hopes that the crisis will stop by itself. "Changing the nature of the entity" whose relations are in crisis is possible, but it rarely happens, as well as "change of circumstances" the wider social and international context in which the conflict or crisis takes place. It is especially difficult to be implemented at the same both changes, which is most advantageous from the point of eliminating the conflict. It remains to compromise is one of the ways to treat some conflicts and crises - as a sublimation of compromise negotiations. Key words: conflicts and crises, diplomacy, diplomatic negotiation, game theory, mathematical modeling

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Introduction Game theory is the mathematical theory of conflict situations. In the field of economy, these situations are called competition. Game theory was created in 1928 when von Neumann proved the theorem on minmax. Later it began to be applied in many other branches of science and life - almost every time you need to develop a strategy in the conflict of interest situations. In the book “Theory of games and Economic Behavior” in 1944, this theory was developed by the economist Oskar Morgenstern and the famous mathematician John von Neumann (Neuman J.von, 1953). They noted that in the economic sciences there are no mathematical models good enough to describe situations in which market participants are facing mutual conflict interests. Therefore, Neumann and Morgenstern described these situations by abstracted games, conceived as a set of rules and conventions that players must follow. At each stage of the game, players pull certain moves by the set of (un) limited decisions making the right choices and choose that decision that seems best to them on the basis of available information. Fundamental contributions to the theory of games were made by John Nash (Nash, 1951). Game theory is the basis for many theories of negotiation used by police and diplomacy. Strategies, developed in the game theory by appropriate mathematical tool, offer players the instruction set for any situation that may arise during a game, and one of the key aspects for making right decision is to consider the possible moves of rival players. In 1994 “for a pioneering analysis of equilibrium in the non-cooperative games theory'' the Nobel Prize in economics received John Nash, John Harnasnhi and Reinhard Zelten. In 2005 for “increasing our understanding of conflict and cooperation through game-theory analysis” "the Nobel Prize in economics received Robert Auman and Thomas Schelling. The paper will discuss the theory of games in the function of modeling of police negotiation for resolution of conflicts situations. The police have a role in resolving conflict situations in the field of security. In doing so, it has a duty to protect also the security of holders of threat. Minimization of harmful effects to the life and health of all participants in the conflict requires the application of police methods in the resolving noncompulsory conflict situations. One of these methods is negotiation. Negotiation is a process of mutual persuasion of opposing sides in the communication in terms of antagonism of their goals. Police (security) negotiation is usually carried out in the cases of kidnapping, hostage situations, severe forms of extortion, riots in prisons, occupation of buildings, street demonstrations, threats of suicide and homicide (murders), threats to the police or third parties by weapons or explosive devices in the 423

preparation and implementation of police measures, resistance to police measures etc., when it is possible to influence the behavior of perpetrators of conflict situations, with a view to their withdrawal from illegal conducting (Subošić, 2010.). Due to the complexity of circumstances in which police negotiation is implemented, it is necessary for its modeling, to predict its effects to the greatest possible certainty. Maximization of police negotiation effects is achieved by optimal strategies through which it realizes. Mathematical (exact) method to do this is the application of game theory. This paper questions the right application of game theory in the modeling function of police negotiation Basic Terms of the Game Theory           

Basic terms of the game theory are: game move strategy optimal strategy pure strategy mixed strategy low value of matrix game high value of matrix game saddle point of matrix game value of game goal of game theory

Game is a model of conflict situation. Move is a selection of one of the possible alternatives available to participants in the game. Strategy is a set of rules that uniquely determine the choice of gait of each participant in the game. Optimal strategy is a kind of strategy that during multiple repeated games provides the participant to achieve the maximum possible medium payoff, i.e., minimum possible medium loss. In the game theory there is already mentioned principle (criterion) of the minimax. It is expressed by the view that a player in the matrix game (conflict situation simulation) chooses his behavior in a way that maximizes his payoff with, for him, the most adverse action of opponent. By this principle the choice of the most cautious strategy for each player is conditioned. At the same time, the minimax is also the basic principle of game theory. Strategies chosen trough this principle are

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called minimax strategies (Petrić, 1988). Pure strategies are at least one strategy that both players have at their disposal, which are predicted to be better than all the strategies of opponents. Mixed strategy is a complex strategy, which consists in applying more pure strategies in a certain respect. This type of strategy can be reached via the selection probability of one pure strategy (р1) and selection probability of other pure strategy (р2 =1-р1). At the same time, the value of р1 is in the range from 0 to 1. The low value of matrix game is the maximum payoff between minimum payoffs (α). The high value of matrix game is the minimal loss between maximal effects (β). The saddle point of matrix game is that point which is expressed by the maximum in its column and by the minimum in its row (Milovanović, 2004.). It exists in the case when the low and high value of matrix game are of identical values (α = β). If α is not equal to β, there are no saddle. The difference between α and β is ''space'' in which the participants in the matrix game should demonstrate own abilities, by choice of optimal - mixed strategy. The value of game (v) is the value between the maximum payoff between minimum payoffs (α) and minimal loss between maximum effects (β). Mathematically expressed, the value of game is the interval:     . The goal of game theory is the determination of optimal strategy for each participant. Classification of Matrix Games Games can be divided on various grounds: the number of players, the number of available strategies, as a function of time, the choice of strategy, according to whether the sum is constant or variable. Division of games according to the number of players According to the number of players, games are divided into games with two, three, …, players. Games with two players are most often used in the game theory, which is not accidental, given the fact that in everyday life, we most often meet this type of games (such as chess, for example). These games are analyzed in detail in the game theory. The aim is to describe the condition by using a very well known form, i.e. by a game whose characteristics are known in advance and are actually predictable. Games with n players are generalizations of games involving two players. Further in the presentation, the emphasis will be on games involving two players.

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Division of games according to the number of available strategies According to the number of available strategies, games are divided into games with a finite and infinite number of strategies. If at least one player has an infinite set of strategies, then the game with the infinite number of strategies is in question. Division of games according to the time function Depending on whether the strategies of the players are in the function of time, we talk about dynamic and static games. If the players make their decisions simultaneously, then we talk about static games. Otherwise, we talk about dynamic games. Division of games according to the choice of strategy The choice of strategy in a game can be deterministic or stochastic. Games in which we have a random choice of strategies (for example, when tossing a dice) are called stochastic games. Games with constant and variable sums A payoff of each player in the game is the result of the strategy chosen by this player, but also the action of all other participants in the game. According to the final outcome (the total payoff), games can be played with a constant or a variable sum. A zero-sum game is a game in which the gain of one participant is identical to the loss of the other participant. All games can be presented in three different ways: extensive form games, normal form games and games in a form of a characteristic function. Extensive form games are those presented in a form of a tree. The way in which we present normal form games gives a matrix wherein the rows represents available actions of the first player, while columns show the actions of the second player. Each of the players has a disposable number of strategies. All disposable actions of one player make his strategic set. It has been understood in this presentation that both players are rational, i.e. that they want to maximize their gains. Matrix game is a game that can be realized by the following rules:  The game involves two players;  Each player has a finite set of available strategies;  The game consists in the fact that each player having no information about the intentions of opponent makes a move (chooses one of the strategies). As a result of the chosen strategy arises payoff or loss in 426

the game;  Both payoff and loss in the game are expressed as number; The strategy of player I will be seen as the row of some matrix, and the strategy of player II as the column of some matrix. The situations in the game are presented by boxes at the intersection of rows and columns. Filled boxes – situations – by real numbers that represent the player’s I payoff, we give a task in the game. Resulting matrix is the winning matrix of game or game matrix. Taking into account the antagonism of matrix game, the payoff of player II in any situation means the loss of player I and differs only by sign. No additional explanation on the function of winning player II is required. The matrix that has m rows and n columns is called (m  n) matrix, and game (m  n) . There are simple and mixed matrix games. They differ in how the simple games have and mixed have not so-called “saddle point”. In addition, the simple matrix games correspond to the situations of certainty, while the mixed situations correspond to the situation of uncertainty (Milovanović, 2004.). Modeling of Police Negotiation through the Matrix Game with Saddle For the games with a saddle is characteristic that they have clear strategies for both players, which by row and column correspond to the saddle point. The example of the police negotiation (see chart below) means that the police (player no. 1) has at its disposal two strategies 1 (x1, x2), while the opposing party (player no. 2) also has at its disposal two strategies (y1, y2). For example, both strategies for both parties are indulgent and unyielding negotiation. Figure 1. Saddle point

1

In general case it means that he has at least two strategies

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Table 1. Player strategy Players Alternatives I x1 II y1

x2 y2

x3 y3

Let's take a concrete example. Game payoff for the player I , which depends on the choice of possible strategies, is given by the following set of data, v(x1,y1) = 4, v (x1,y2) = - 1, v (x1,y3) = -4, v (x2,y1) = 3, v (x2,y2) = 2 (4.1) v (x3,y1) = -2, v (x3,y2) = 0 v (x3,y3) = 8. Find a solution to a game, i.e. determine: a) optimal strategic pair (xi, yj), b) find the value of matrix game

v (x2,y3) = 3,

Resolution: The game defined above can be reduced to the matrix form, as shown in Table 2, where rows correspond to the possible strategies of player I and column to the possible strategies of player II. Table 2.

Game with saddle point Player strategy II

Player strategy I

Minimum by rows

y1

y2

y3

x1

4

-1

-4

-4

x2 x3

3 -2

2 0

3 8

2 -2

4

2

8

Maximum by column

By analysis of the matrix game price, the player I determines that if he chooses strategy x1 at least he will get is -4, for strategy x2 is 2 and if he chooses strategy x3 minimum payoff is -2. The player I will try to choose such a strategy which corresponds to the maximum among specified minimum payoffs. In our case, it is strategy x2. The payoff value, which corresponds to strategy x2, is called the low value of game and is marked with α. Therefore, we have that  α = maxi minj (aij) = 2 (4.2) By analysis of the matrix game price, the player II determines that if he chooses strategy y1 the maximum what he can lose is 4, for the strategy y2 is 2, and if he chooses strategy y3 the maximum that he can lose is 8. The player II will try to choose such a strategy corresponding to the minimum of

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specified maximum losses in each column. In our case it is strategy y2. Thus, the value obtained is called the high value of game and is marked with β. Therefore, we have that  β = minj maxi (aij) = 2 (4.3) If the high value of game is equal to the low value of game for such matrix game is said to have a saddle, and solution to the game is in the domain of pure strategies. In other words, if both players find at least one strategy that is the best according to prediction in relation to all the strategies of his opponent it is said that the game has as solution the pure strategies as optimal. This is possible only if the matrix game has a saddle. In this case, the value of game is 

v

=α=β.

(4.4)

  In our case, the matrix game has the saddle and optimal strategies are in the domain of pure strategies, namely:  I player - strategy x2,  II player - strategy y2,   and the low value of game is equal to the high value of game, i.e. v = α = β = 2. (4.5) The element a22 = 2 in the matrix aij is called the saddle of matrix game. If the player I applies any other strategy, not the strategy x2, and player II remains at the optimal strategy, the payoff of the player I will be reduced. Also, if the player II tries any other strategy and not y2, and the player I maintains its optimal strategy, the loss of player II will be increased (Petrić, 1988.). Police Negotiation Modeling through Mixed Matrix Game Mixed matrix games are divided into: matrix game (2  2) , matrix game (n  2) , matrix game (2  m) and (n  m) . Solving the matrix games (n  2) and (2  m) is conducted by reducing them to the matrix game (2  2) . Solving the matrix games n x m can be done by reducing the price matrix and linear programming. Finally, in addition to the above, there are also mixed tasks. Mixed matrix games (2  2) are the matrix games types characterized by existence of two pure strategies of each participant (police negotiator and police opponent), and there is no'' saddle'', which is why their solution lies in

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the field of mixed strategies. Solving the matrix game is shown in the following example. While the party I (police) calls a dangerous criminal whose arrest is in progress, to surrender, the party II (dangerous criminal) refuses to surrender. In order to minimize harmful effects to the life or health in this conflicting situation, there comes to negotiation. The task is: find the low and high value of matrix game (α and β), determine the optimal strategies of participants and the value of game (P, Q, v). The probabilities for all combinations of opposing parties’ strategies are given in the following table. Table 3. The probabilities for all combinations strategies y1 y2 x1 0,4 0,2 x2 0,2 0,6 Maximum by β = 0,4 0,6 column

of opposing parties Minimum by row α = 0,2 0,2

The high and low value of matrix game has no identical values (α ≠ β), so that the matrix game does not have: “saddle” It also means that the matrix game has an optimal strategy in the mixed strategies domains. When the payment matrix has no saddle point, then it is somehow more difficult to determine the optimal strategies of players and the value of matrix games. Thus the Player I does not have a clear strategy that would provide a guaranteed minimum gain, along with the rational behavior of the other player. By analogy, the Player II does not have a strategy that would provide the upper limit of his payments. Therefore, the players introduce the element of randomness in the selection of particular strategies. They no longer choose one strategy, but opt for various strategies. Each strategy appears with certain likelihood. By using the von Neumann theorem, optimal strategies can be determined in all zero-sum games (Subosic, Stevanovic, Jacimovski, 2013). Game Theory in the Function of Negotiations This paper deals with the game theory that is used for the analysis of interactions that involve negotiations. We use the term “negotiations" to describe a situation in which: individuals (players) have the possibility of reaching an agreement, there is a conflict of interests around which an agreement is to be achieved,

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

it is not possible to impose an agreement to which either party has not given its consent.

A negotiation theory represents the examination of relationships between the outcome of negotiations and the circumstances. We assume that the players are rational and have equal negotiation skills, that both parties (or as many as they are) have clearly defined priorities for each possible outcome and should the player make a choice, he will choose the alternative that will lead him to the best possible outcome. We shall try to impose a certain negotiation structure and study the outcomes that are covered by the notion of a perfect equilibrium. The structure that we are going to impose makes the players be maximally symmetrical. Player I comes with the offer that Player II can either accept or reject. In case of rejection, Player II gives his offer that Player I can either accept or reject, etc. Almost all the models have a sequential structure: the players must make decisions one after the other, according to a pre-determined order. This order reflects the progress of negotiations and the process of trade. The time needed for reaching an agreement is also of importance to the negotiators. Such a structure is flexible and allows us to solve a wide range of issues. Two rational individuals, I and II, voluntarily enter into negotiations. They know all the possible outcomes of the negotiations, to which they have attributed various benefits (based on their preferences). Negotiations can be completed with different outcomes, and a set of all results that individuals can achieve represents a so called accessible area of D (Figure 2), (Pavlicic, 2000).

Negotiating set of P

Figure 2. Accessible area 431

Accessible area contains the point of failure (status quo, the point of misunderstanding), which shows the downturn of the agreement. The benefits of the negotiators at the point of failure are marked with N = (n 1 , n 2 ). Other points in the accessible area represent possible agreements that are acceptable in different ways for both individuals. Since they are perfectly rational, negotiators will not observe all the possible solutions, but only those that are Pareto optimal1 (in which both negotiators achieve better outcomes compared to inefficient solutions). The set of all Pareto optimal points makes a so called negotiating set of P (shown in the figure by a subset of limit points of the accessible areas). Points of the negotiating set are shown by using utility pairs (x, y). It is clear that individuals prefer any point in the set of P in relation to the point of failure, so there is a mutual desire to reach an agreement. In case of an agreement, they make gains that are equal to: (x - n 1 ) for the first and (y- n 2 ) for the second individual. If the set of P contains only one point, the problem becomes trivial and the agreement will be concluded immediately. Therefore, we assume that the set of P contains at least two points that individuals prefer in a different manner, which again raises the problem of how to predict the point of their agreement (x*, y*). Nash has set the following conditions that any solution of the negotiations should satisfy (Mucibabic, 2006):  efficiency of a solution with regard to Pareto,  symmetry,  invariance in relation to the linear transformation of the utility function, and  independence from irrelevant alternatives. Starting from the mentioned four assumptions, Nash has proved the following theorem: If the conditions 1-4 have been met, then the point of agreement (x*, y*) represents the solution of the function max (x-n 1 )(y-n 2 ) within the negotiating set, on condition that there are such points that make valid the functions x>n 1 and y>n 2 . The obtained solution is called Nash equilibrium and it is the only solution that satisfies all four conditions. A geometric illustration of Nash solution is shown in the figure 3 (Mucibabic, 2006):

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