In experiments designed to uncover the neural basis of adaptive decision making in a foraging environment, neuroscientists have reported single-cell activities in the lateral intraparietal cortex (LIP) that are correlated with choice options and their subjective values. To investigate the underlying synaptic mechanism, we considered a spiking neuron model of decision making endowed with synaptic plasticity that follows a reward-dependent stochastic Hebbian learning rule. This general model is tested in a matching task in which rewards on two targets are scheduled randomly with different rates. Our main results are threefold. First, we show that plastic synapses provide a natural way to integrate past rewards and estimate the local (in time) "return" of a choice. Second, our model reproduces the matching behavior (i.e., the proportional allocation of choices matches the relative reinforcement obtained on those choices, which is achieved through melioration in individual trials). Our model also explains the observed "undermatching" phenomenon and points to biophysical constraints (such as finite learning rate and stochastic neuronal firing) that set the limits to matching behavior. Third, although our decision model is an attractor network exhibiting winner-take-all competition, it captures graded neural spiking activities observed in LIP, when the latter were sorted according to the choices and the difference in the returns for the two targets. These results suggest that neurons in LIP are involved in selecting the oculomotor responses, whereas rewards are integrated and stored elsewhere, possibly by plastic synapses and in the form of the return rather than income of choice options.