Auction design for the modern advertising market has gained significant prominence in the field of game theory. With the recent rise of auto-bidding tools, an increasing number of advertisers in the market are utilizing these tools for auctions. The diverse array of auto-bidding tools has made auction design more challenging. Various types of bidders, such as quasi-linear utility maximizers and constrained value maximizers, coexist within this dynamic gaming environment. We study sponsored search auction design in such a mixed-bidder world and aim to design truthful mechanisms that maximize the total social welfare. To simultaneously capture the classical utility and the value-max utility, we introduce an allowance utility model. In this model, each bidder is endowed with an additional allowance parameter, signifying the threshold up to which the bidder can maintain a value-max strategy. The paper distinguishes two settings based on the accessibility of the allowance information. In the case where each bidder’s allowance is public, we demonstrate the existence of a truthful mechanism achieving an approximation ratio of \((1+\epsilon )\) for any \(\epsilon > 0\) . In the more challenging private allowance setting, we establish that a truthful mechanism can achieve a constant approximation. Further, we consider uniform-price auction design in large markets and give a truthful mechanism that sets a uniform price in a random manner and admits bounded approximation in expectation.