**2. Theory and literature review**

Innovation is central to how firms compete over time; "firms that succeed in innovation prosper, at the expense of their less able competitors" ([2], p 20). This has led to extensive interest in how firms manage innovation (e.g., [3–5]) and particularly on how they can use external sources (or inbound innovation) to conduct open innovation ([6–9], and many more). Much of this open innovation work, though, has come in the form of case studies, typically covering small geographical regions, across a small number of industries or firms, and focusing on specific areas of inquiry [6, 7, 10–12]. As West et al. [13] describes these studies "provide high internal validity but offer limited external validity: as such, they are better at 'why' or 'how' questions than 'when' or 'how often'." Our paper presents a broad international view on 'when' and 'how often', adding to the existing literature in this area that has mainly focused on European firms, through the Community Innovation survey (CIS) [14–19], and the U. S. manufacturing sector [20, 21]. There have also been much-shorter, non-representative surveys that we do not consider here (e.g., [22] with a completion rate of <5%, consulting reports, etc.).

Our work makes several contributions beyond the existing literature. First, our survey provides a more representative view of global corporate innovation by offering greater geographic and industry reach than the CIS or US manufacturing surveys. Second, our survey is more current, reporting data from 2018. The other "how often" papers listed earlier report data from predominantly the 1990s, with the latest from 2010 from Arora et al. [20] (hereafter ACW). This almost-decade gap matters because, as our results show, much of the popularization of the open innovation revolution has only happened in recent years and this has been accompanied by a rapid shift in the usage of innovation sources. A more up-to-date view of corporate innovation is also important for understanding digital innovation, since the period from 1980 to 2019 has been accompanied by an enormous escalation in digital investment by firms, and thus earlier studies would underestimate the role that digital innovation is playing today. To put the size of this shift in context, US firm spending on business software over this period rose from 5 to 33% of equipment spend [23]. The post-2010 period is also when firm investment in predictive analytics becomes a key driver of productivity gains [24] and when digital co-design, for example the Open Compute Project launched in 2011, comes to the fore [25].

A third advantage of our survey is that it is more managerially relevant because we take a firm-centric definition of innovation (rather than industry-centric one [1]. This more accurately represents the decisions facing innovation managers, since their remit is broader than just the new-to-the-industry discoveries covered by industrylevel analyses, and thus such analyses miss an important part of real managerial decisions. Indeed, testing in one of our geographies reveals that only 27% of firms' most successful innovations came from outside the industry. A firm-centric approach is also likely to do a better job capturing the innovations that drive overall productivity growth, since intra-industry differences also represent the bulk of variation in firm performance [26] and productivity [27]. Consistent with this claim, testing in one geography reveals that innovation projects that are new to the industry provide competitive advantage at almost exactly the same rate as innovations that are not new to the industry (we are able to test this difference because we tested the addition of this question in one geography). And thus the industry-level approach pays a high

price in omitting projects that are important to firms and provide them competitive advantage. The firm-level approach does, however, come at a cost of our ability to aggregate. Whereas other papers can make a claim about the share of innovation coming into an industry from outside of it, we cannot make such statements.
