## Abstract

Web-based Doodle polls, where respondents indicate their availability for a collection of times provided by the poll initiator, are an increasingly common way of selecting a time for an event or meeting. Yet group dynamics can markedly influence an individual’s response, and thus the overall solution quality. Via theoretical worst-case analysis, we analyze certain common behaviors of Doodle poll respondents, including when participants are either more generous with or more protective of their time, showing that deviating from one’s “true availability” can have a substantial impact on the overall quality of the selected time. We show perhaps counter-intuitively that being more generous with your time can lead to inferior time slots being selected, and being more protective of your time can lead to superior time slots being selected. We also bound the improvement and degradation of outcome quality under both types of behaviors.

### Keywords

- Social Welfare
- Time Slot
- Vote Behavior
- Strategic Vote
- Approval Vote

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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## A Appendix

### A Appendix

### 1.1 A.1 Restrictive Voting

### Lemma 14

Even if the score of the winning slot must be at least *n* / 2 and there is only one restrictive voter, the positive welfare impact factor of restrictive voting is still at least \(\frac{1}{t} + 1\) (suppressing epsilons).

### Proof

Consider the instance represented in Table 2b. The utilities of the time slots are \(u(a_1) = nt/2\) and \(u(a_2) = n/2 + n(t-\epsilon )/2\). When all participants vote according to the yes-threshold *t*, \(s^*(a_1) = n/2 = s^*(a_2)\), with the tie meaning that slot 1 could be chosen. When the one restrictive voter votes restrictively, still saying yes to slot 2 but now saying no to slot 1, the reported scores become \(s'(a_1) = n/2 -1\) and \(s'(a_2) = n/2\), so that slot 2 is now chosen. Thus, suppressing epsilons, the social welfare improves by a factor of \(1/t + 1\).

### Lemma 15

Even if the score of the winning slot is *n* and there is only one restrictive voter, the positive welfare impact factor of restrictive voting is still at least \(\frac{1}{t}\).

### Proof

Consider the instance represented in Table 2c. The utilities of the time slots are \(u(a_1) = n\) and \(u(a_2) = nt\). When all participants vote according to the yes-threshold *t*, \(s^*(a_1) = n = s^*(a_2)\), with the tie meaning that slot \(a_2\) could be chosen. When the one restrictive voter votes restrictively, still saying yes to slot 1 but now saying no to slot 2, the reported scores become \(s'(a_1) = n\) and \(s'(a_2) = n-1\), so that slot 1 is now chosen. Thus, the social welfare improves by a factor of 1 / *t*.

### Proof

*(of Lemma* 5
*)*. Mirroring the proof of Lemma 3, define \(a=DDL(I',R_O)\) to be the slot selected when everyone votes according to the yes-threshold, and *b* is the slot selected when the \(\ell \) restrictive voters vote restrictively. Since we are analyzing the negative welfare impact factor, we must upper bound *u*(*a*) / *u*(*b*). Observe that by the definitions of *a* and *b* and that since restrictive voting can only lower reported scores, we have that \(s'(a) \le s'(b) \le s^*(b) \le s^*(a)\). With that observation, and noting that, similarly to Fact 1, \(u(b) \ge s^*(b)t\), we then have that \(u(b) \ge s'(a) t\). Since \(t<t'\) in restrictive voting, and a restrictive yes vote indicates a valuation of at most 1, while a restrictive no vote indicates a valuation less than \(t'\), Fact 2 now becomes \(u(a) < s'(a) + (n-s'(a))t'\). A comparable remaining argument to that of Lemma 3 thus gives the resulting upper bound.

Consider the instance represented in Table 6. Let \(k>2\) be a fixed constant, and without loss of generality, assume \(k\mid n\) and \(2\mid n\), for ease of analysis. The instance has *n* / 2 restrictive voters with valuations as before, but also *n* / 2 ordinary voters, numbered \(i=1\) to *n* / 2, who all value slot \(n/2+1\) at \(t-\epsilon \), the slots *i* to \(i + n/k - 1\) (wrapping around for slots exceeding *n* / 2) at *t*, and the rest at 0. The utilities of the time slots are \(u(a_1) = \ldots = u(a_{n/2}) = ((n/k)+1)t+\epsilon \) and \(u(a_{n/2+1}) = nt - n\epsilon /2\). When all participants vote according to the yes-threshold *t*, the reported scores are \(s^*(a_1) = \cdots = s^*(a_{n/2}) = n/k + 1\) and \(s^*(a_{n/2+1}) = n/2\). When the restrictive voters vote restrictively, saying yes to their one slot with valuation \(t + \epsilon \) but no to the slot with valuation *t*, the reported scores are \(s'(a_1) = \cdots = s'(a_{n/2}) = n/k + 1\) and \(s^*(a_{n/2+1}) =0\). Thus, since \(k>2\), restrictive voting changes the selected time slot from slot \(n/2+1\) to any of the other slots, decreasing the social welfare from *nt* to \((n/k + 1)t\), suppressing epsilons. Note that while this example does have a more plausible reported score, it does require the number of time slots to be about half of the number of participants.

### Lemma 16

The negative welfare impact factor of restrictive voting is at least \(\approx m\).

### Proof

Consider the instance represented in Table 7. The last slot has utility \(u(a_m) = nt\), while the other slots have utilities \(\lfloor \sqrt{n} \rfloor (t+\epsilon )\), except possibly for slot \(m-1\) which may have smaller utility, due to the square root and truncation with the floor operation. When all participants vote according to the yes-threshold *t*, most of the slots likewise have reported score \(\lfloor \sqrt{n} \rfloor \), again with slot \(m-1\) possibly lower, and slot *m* having \(s^*(a_m) = n\). When all *n* restrictive voters vote restrictively, that is, no to slot *m*, the reported scores of the first \(m-1\) slots are unchanged, with most at \(\lfloor \sqrt{n} \rfloor \), but slot *m* now has \(s'(a_m) = 0\). Thus, restrictive voting changes the selected time slot from slot *m* to one of the earlier ones (except perhaps for \(m-1\)), decreasing the social welfare from *nt* to \(\lfloor \sqrt{n} \rfloor (t+\epsilon )\), giving the desired result.

Similarly, the instance in Table 7 with restrictive voting can be transformed to an instance showing that cooperative voting can improve social welfare by a factor of \(\approx m\) by making all voters cooperative, changing valuations of *t* to \(t-\epsilon \), and valuations of \(t+\epsilon \) to *t*.

### 1.2 A.2 Cooperative Voting

We again define \(s^*(a)=s(DDL(I',R_O))\) to be the score of the winning slot in an instance \(I'\) when all voters are ordinary. We now let \(s'(a)\) indicate the reported score of slot *a* when the cooperative voters use an adjusted yes-threshold \(t'<t\).

### Proof

*(of Lemma* 8
*)*. Consider the instance represented in Table 4a. The utilities of the time slots are \(u(a_1) = \cdots = u(a_n) = t\) and \(u(a_{n+1}) = c(t-\epsilon )\). When all participants vote according to the yes-threshold *t*, the reported scores are \(s^*(a_1) = \cdots = s^*(a_n) = 1\) and \(s^*(a_{n+1}) = 0\). When the *c* cooperative voters vote cooperatively, the reported scores become \(s'(a_1) = \cdots = s'(a_n) = 1\) and \(s'(a_{n+1}) = c\). Thus, cooperative voting changes the selected time slot from any of the first *n* to time slot \(n+1\), increasing the social welfare by a factor of *c*, suppressing epsilons.

### Proof

*(of Lemma* 9
*)*. Consider the instance represented in Table 4b. The utilities of the time slots are \(u(a_1) = c(t-\epsilon ) + n - c\), and \(u(a_2) = (n-c)t\). When all participants vote according to the yes-threshold *t*, the first group of *c* participants report no for both slots, while the second group report yes for both slots. Thus, \(s^*(a_1) = n - c = s^*(a_2)\), with the tie meaning either slot can be chosen. When the *c* cooperative voters vote cooperatively, they vote yes for slot 1 but still no on slot 2. The ordinary voters are unchanged in their votes. Hence, \(s'(a_1) = n\) and \(s'(a_2) = n-c\), ensuring that slot \(a_1\) is chosen. The improvement in social welfare when slot \(a_1\) is chosen due to cooperative voters rather than when slot \(a_2\) can be chosen when all voters vote ordinarily is thus a factor of \(\frac{1}{t}+\frac{c}{n-c}\) (suppressing epsilon terms).

### Proof

*(of Lemma* 10
*)*. The proof is analogous to the proof of Lemma 3, except rather than \(\ell \) restrictive voters, we have *c* cooperative voters. Note that Facts 1 and 2 which lower bound the utility of slot *a*, the slot that is chosen when everyone is an ordinary voter, and upper bound the utility of slot *b*, the slot that is chosen when *c* of the *n* voters vote cooperatively, still stand as they are established purely on the reported scores of the two time slots when all voters are ordinary. We therefore still have the established upper bound on the welfare approximation ratio of

### Proof

*(of Lemma* 11
*)*. Consider the instance represented in Table 5b. If we set \(t'=\epsilon \), the utilities of the time slots are \(u(a_1) = n/2\) and \(u(a_2) = n\epsilon /2\). When all participants vote according to the yes-threshold *t*, the reported scores are \(s^*(a_1) = n/2\) and \(s^*(a_2) = 0\). When the first group (half of the participants) vote cooperatively, the reported scores are \(s'(a_1) = n/2\) and \(s'(a_2) = n/2\). Thus, with cooperative voting, slot \(a_2\) may be chosen instead of \(a_1\). Hence, the utility goes from *n* / 2 to \(n\epsilon /2\).

### Proof

*(of Lemma* 12
*)*. Consider the instance represented in Table 5c. The utilities of the slots are \(u(a_1) = ct'\) and \(u(a_2) = c+(n-c)(t-\epsilon )\). When all participants vote according to the yes-threshold *t*, the reported scores are \(s^*(a_1) = 0\) and \(s^*(a_2) = c\). When the *c* cooperative voters vote cooperatively, the reported scores become \(s'(a_1) = c\) and \(s'(a_2) = c\). Thus, with cooperative voting, slot 1 may be chosen instead of slot 2, resulting in the indicated change in social welfare.

### Proof

*(of Lemma* 13
*)*. Mirroring the proof of Lemma 3, define \(a=DDL(I',R_O)\) to be the slot selected when everyone votes according to the yes-threshold, and *b* is the slot selected when the *c* cooperative voters vote cooperatively. Since we are analyzing the negative welfare impact factor, we must upper bound *u*(*a*) / *u*(*b*). Observe that since cooperative voters have a lowered threshold of \(t'\), the claim paralleling Fact 1 is \(u(b) \ge s'(b)t'\). We also know that \(s'(b)\ge s^*(a)\) since with cooperative voting there can only be more yes votes than under ordinary voting, so the winning score of *b* must be at least that of *a*. Taking these two inequalities together gives us \(u(b) \ge s^*(a)t'\). Fact 2 now becomes \(u(a) < s^*(a) + (n-s^*(a))t\). A comparable argument to that in the restrictive voting section thus gives the resulting upper bound.

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Alrawi, D., Anthony, B.M., Chung, C. (2016). How Well Do Doodle Polls Do?. In: Spiro, E., Ahn, YY. (eds) Social Informatics. SocInfo 2016. Lecture Notes in Computer Science(), vol 10046. Springer, Cham. https://doi.org/10.1007/978-3-319-47880-7_1

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