Vying for Allah’s Vote: Understanding Islamic Parties, Political Violence, and Extremism in Pakistan

Appendix 1

APPENDIX 1 T he sharia-secular model is an adaptation of a classic two-player signaling game. Parties 1 and 2 are both preparing for an upcoming election. The voters know that each party prefers either constituency A or constituency B, but they do not know where the parties’ real preferences lie. The probability that a party actually prefers constituency B is r, meaning that a party actually prefers constituency A with probability 1 ⳮ r. Values for r are not provided here, but it might be reasonable to assume that r is equivalent to the proportion of voters favoring constituency B. Party 1, as mentioned previously, will always attempt to pass legislation favoring constituency A; therefore, the upper left path, in which party 1 courts constituency A by working to pass favorable legislation, is a strictly dominant strategy both before and after the election. Party 2, however, will have to choose between espousing a policy that favors the majority constituency (including working in the run-up to the vote to pass legislation to prove its commitment) (selected with probability p) and one that favors its true constituency (selected with probability 1 ⳮ p). Constituency A voters, therefore, face a dilemma when p ⬎ 0 (when p ⳱ 0, there is a separating equilibrium). Given that both parties are sending the same signals, they must decide how likely it is that either party will truly support their interests when in office. A party may be lying (probability q) or telling the truth (probability 1 ⳮ q). Figure A.1 represents the first stage of the game: the first elections. The terminal nodes are labeled with payoffs for parties (top) and constituency A (bottom). The payoff variables are defined as follows, with Y ⬎ F and S ⬎ N ⬎ R. There are three equilibrium in stage 1: partially separating, pooling, and separating . We solve first for the partially separating equilibrium, where party 2 mixes strategies (that is, neither lies all the time nor tells the truth all the time). (0.1) q⳱ pr prⳭ1(1ⳮr) 189 190 APPENDIX 1 Figure A.1. Sharia-secular model 1 1 – r r Party 1 Party 2 Pro-Const. A policy Pro-Const. A policy Pro-Const. B policy Pro-Const. B policy 1 – p p q 1 – q Const. A favors Const. A favors Const. A disfavors Const. A disfavors Const. A disfavors Const. A disfavors Const. A favors Const. A favors Y S + G F N + G F – Z N + G Y – Z R + G Y R F N Stage 1 DOMINATED Table A.1. Payoff variable definitions Y—payoff to winning party G—benefit to constituency A if policy passed in its favor prior to election F—payoff to losing party R—payoff to constituency A if it elects party 1 Z—cost to party 2 of supporting constituency A S—payoff to constituency A if it elects party 2 N—payoff to constituency A if it abstains Rearranging, we get: (0.2) p⳱ 1ⳮr r 冉 q 1ⳮq 冊 Now, set the payoffs to constituency A for supporting a party favoring its policy equal to the payoffs for denying support to such a party. (0.3) q(RⳭG)Ⳮ(1ⳮq)(SⳭG)⳱q(NⳭG)Ⳮ(1ⳮq)(NⳭG) This equation reduces to: (0.4) q*⳱ SⳮN SⳮR APPENDIX 1 191 Plugging (0.4) into (0.2), we get the optimal mixing probability of party 2, the optimal percentage of the time that party 2 will try to attract constituency A’s support: (0.5) p*⳱ 1ⳮr r 冉SⳮN NⳮR 冊 Next, we look at the pooling equilibrium, where both parties always support policies favoring constituency A. In this case, p* ⳱ 1, so q* ⳱ r*. We are left with: (0.6) r*⳱ SⳮN SⳮR If r ⳱ r*, constituency A is indifferent between supporting and not supporting a party that signals favorable policies before the election. If r ⬎ r*, constituency A will decide not to support either party. This decision will lead party 2 to defect and signal a strategy favoring constituency B. If r ⬍ r*, constituency A will always support a party that signals a favorable policy. Thus, party 2 will only defect when the payoff to losing (with no constituency A support) exceeds the payoff to supporting constituency A and winning. In mathematical terms: (0.7) F ⬎YⳮZ or Z ⬎ YⳮF For a separating...