We consider two prominent mechanisms for the electricity market; the pay-as-bid mechanism, currently applied in certain control reserve markets, and the proposed Vickrey- Clarke-Groves mechanism, an established auction mechanism used in advertising and spectrum auctions, for example. Bringing in tools from game theory and auction theory, we compare the Nash equilibria of these two mechanisms in terms of social efficiency and strategic behavior of the players. Furthermore, by formulating a coalitional game corresponding to the electricity market, we propose alternative mechanisms that incentivize truthful bidding while ensuring shill bidding is not profitable. Finally, we analyze the proposed mechanisms in a case study based on electricity market data.