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Confrontation analysis

Simulated Expected Utility Calculation

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Calculo de probabilidades

2.4. Probability of Success (P)

The probability of success for player i in competition with player j is also calculated by the support of third-party players for player i’s policies versus player j’s policies. Similar to finding the median voter position,

it is not only about which player’s policies the parties prefer, but also the third-parties’ salience on the issue and their capability or power are considered.

Equation (4) shows the probability of success for player i in competition with player j according to Expected Utility Model [3].

xxxxx

The numerator calculates the expected level of support for i. The denominator calculates the sum of the

support for i and for j so that the expression shows the probability of success for i, and it obviously falls in the

range of 0 and 1.

2.5. Probability of Status Quo (Q)

The calculation of the probability of status quo (Q) has never been talked about in any of Bruce Beuno de

Mesquita’s publications. He considers Q constant in some publications such as [9] in which he considers Q to be

1. This means that when i decides not to challenge j, i and j remain in stalemate with each other and the change

in their positions in responding to other players’ offers does not affect their situation against each other. In

different publications [10] and [11], he considers Q to be 0.5 which means whether the change in the position

affects players situations versus one another or not, is completely random. In this work, we have been able to

calculate this probability for each pair of players. According to Figure 2, when A does not challenge B, B is

challenged by other players and may lose and be forced to move. If B moves, its position changes and its

distance to A either decreases (with probability T) or increases (with probability 1-T). Therefore, the probability

of status quo in this situation is the probability that B does not move at all. This is the probability that B wins the

challenge with every other player except A in that round. This probability is calculted as follows:

The probability that player(i) wins every challenge against another player(k), is the sum of two probabilities.

first, the probability that player(i) challenges player(k) and player k does not challenge it back and surrenders

which is (1 − Sk ) . Second, the probability that player(i) challenges player(k) and player(k) does respond to its

challenge and again player(i) wins this confrontation which is Pi . The probability that player(i) wins against

every other player except player(j), is the multiplication of this sum for all players except player(j).

2.6. Probability of Positive Change (T)

According to Figure 2, when A decides not to challenge B, but B moves due to other challenges, its move either

improves or worsens the situation for A. B move would be towards the median voter position, so the positions of

A, B and the median voter ( µ ) versus one another determines whether B moves closer to A or further away

from it. If B moves closer to A, it improves the situation for A, so T = 1. If B moves further away from A, it

worsens the situation for A, so T = 0 [3].

3.1. Expected Utility Model’s Risk-Taking Component

Expected Utility Model’s risk taking component is explained in details in Bruce Beuno de Mesquita and

Stokman [12]. The risk-taking component is a trade off between political satisfaction and policy satisfaction [13].

Political satisfaction or security is being seen as a member of winning coalition while policy satisfaction is

supporting the policy that is most close to that of the player itself even if that policy does not win. The rate at

which players make this trade-off is different from one another. The security of a player increases and the risk

decreases by taking a position close to the median voter position. Therefore, the players who take positions close

to the median voter position, who is the winner at the associated round, are feeling more vulnerable and tend to

be more risk averse [5]. What enters the calculation of risk in the Expected Utility Model, is the actual expected

utility, the maximum feasible expected utility and the minimum feasible expected utility. Algebraically, the

risk-taking component is calculated as follows [9]:

ijmax

ini

ijmin

ini

∑

i=1

ijmax

1−

2∑

− ∑i=1

ij −∑i=1

−∑i=1

(5)

(6)

and

As seen in Equation (3), the risk factor is used in the calculation of expected utilities, and according to

Equation (5), risk is calculated using the expected utilities. It can be interpreted from Bruce Beuno de

Mesquita’s statements that first the expected utilities are calculated without considering the risk (r = 1) and then

these utilities are used to calculate the risk for each player [3].

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