TY - JOUR
T1 - Trigger strategy equilibriums in stochastic differential games with information time lags
T2 - An analysis of cooperative production strategies
AU - Yao, T.
AU - Xu, Susan H.
AU - Jiang, Bin
PY - 2011
Y1 - 2011
N2 - In the current literature of differential games, most studies formulate optimal strategies in feedback (Markovian) equilibriums and ignore repeated interactions among players. In many real-world settings, however, the competitors' action history may have impacts on a firm's decisions. This paper considers the production strategies for several competing firms in an oligopolistic industry. A firm's profit is determined by a continuous-time stochastic demand shock process together with the production strategies of all firms in the industry. A firm's decision is only observable by other firms after an information time lag, induced by the production lead time. We study a history-dependent trigger strategy, whereby firms adopt the cooperative strategy until a firm fails to do so, and thereafter punish the deviating firm by applying the noncooperative equilibrium. We show that, as long as the information time lag is less than a threshold, the trigger strategy is both a Nash equilibrium and a Pareto optimum. We obtain the analytical solutions to the threshold and investigate how the threshold is affected by market growth rate, market volatility, the number of competitors in the industry, and the risk-free rate. Moreover, we investigate the repeated games in a continuous-time setting and provide a tractable approach to derive the trigger-type repeated equilibrium in a Nash-Cournot framework. While the derivation of equilibrium strategies in a stochastic continuous-time setting can be quite challenging, we obtain a solution that is not only analytically simple but also practically applicable.
AB - In the current literature of differential games, most studies formulate optimal strategies in feedback (Markovian) equilibriums and ignore repeated interactions among players. In many real-world settings, however, the competitors' action history may have impacts on a firm's decisions. This paper considers the production strategies for several competing firms in an oligopolistic industry. A firm's profit is determined by a continuous-time stochastic demand shock process together with the production strategies of all firms in the industry. A firm's decision is only observable by other firms after an information time lag, induced by the production lead time. We study a history-dependent trigger strategy, whereby firms adopt the cooperative strategy until a firm fails to do so, and thereafter punish the deviating firm by applying the noncooperative equilibrium. We show that, as long as the information time lag is less than a threshold, the trigger strategy is both a Nash equilibrium and a Pareto optimum. We obtain the analytical solutions to the threshold and investigate how the threshold is affected by market growth rate, market volatility, the number of competitors in the industry, and the risk-free rate. Moreover, we investigate the repeated games in a continuous-time setting and provide a tractable approach to derive the trigger-type repeated equilibrium in a Nash-Cournot framework. While the derivation of equilibrium strategies in a stochastic continuous-time setting can be quite challenging, we obtain a solution that is not only analytically simple but also practically applicable.
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U2 - 10.1137/100819151
DO - 10.1137/100819151
M3 - Article
AN - SCOPUS:79960367625
SN - 0363-0129
VL - 49
SP - 1197
EP - 1220
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 3
ER -