In pliable index coding, we consider a server with $m$ messages and $n$ clients where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most $\mathcal{O}(\log^2(n))$ broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case $\mathcal{O}(n)$ transmissions. In this work, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by $\mathcal{O}(\log^2(n))$ broadcast transmissions. We extend our algorithm to the $t$-requests case, where each client requires $t$ unknown messages that she does not have, and show that our algorithm requires at most $\mathcal{O}(t\log(n)+\log^2(n))$ broadcast transmissions. We construct lower bound instances that require at least $Ω(\log(n))$ transmissions for linear pliable index coding and at least $Ω(t+\log(n))$ transmissions for the $t$-requests case, indicating that both our upper and lower bounds are polynomials of $\log(n)$ and differ within a factor of $\mathcal{O}(\log(n))$. Finally, we provide a probabilistic analysis and show that the required number of transmissions is almost surely $Θ(\log(n))$, as compared to $Θ(n/\log(n))$ for index coding. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to $50\%$ less transmissions.